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Continuous function

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle" style="padding-bottom:0.25em;"><a href="Calculus" title="Calculus">Calculus</a></th></tr><tr><td class="sidebar-image"><big><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}">

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<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}</annotation>

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</math><img src="./17d063dc86a53a2efb1fe86f4a5d47d498652766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.228ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}" loading="lazy"></big></td></tr><tr><td class="sidebar-above" style="padding:0.15em 0.25em 0.3em;font-weight:normal;">

<ul><li><a href="Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem</a></li></ul>

<ul><li><a href="Limit_of_a_function" title="Limit of a function">Limits</a></li>

</ul>

</div><div class="hlist">

<ul><li><a href="Rolle's_theorem" title="Rolle's theorem">Rolle's theorem</a></li>

<li><a href="Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li>

<li><a href="Inverse_function_theorem" title="Inverse function theorem">Inverse function theorem</a></li></ul>

</div></td></tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base);display:block;margin-top:0.65em;"><a href="Differential_calculus" title="Differential calculus">Differential</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading">

Definitions</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Derivative" title="Derivative">Derivative</a> (<a href="Generalizations_of_the_derivative" title="Generalizations of the derivative">generalizations</a>)</li>

<li><a href="Differential_(mathematics)" title="Differential (mathematics)">Differential</a>

<ul><li><a href="Differential_(infinitesimal)" class="mw-redirect" title="Differential (infinitesimal)">infinitesimal</a></li>

<li><a href="Differential_of_a_function" title="Differential of a function">of a function</a></li>

<li><a href="Differential_of_a_function#Differentials_in_several_variables" title="Differential of a function">total</a></li></ul></li></ul></td>

</tr><tr><th class="sidebar-heading">

Concepts</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Notation_for_differentiation" title="Notation for differentiation">Differentiation notation</a></li>

<li><a href="Second_derivative" title="Second derivative">Second derivative</a></li>

<li><a href="Implicit_function" title="Implicit function">Implicit differentiation</a></li>

<li><a href="Logarithmic_differentiation" title="Logarithmic differentiation">Logarithmic differentiation</a></li>

<li><a href="Related_rates" title="Related rates">Related rates</a></li>

<li><a href="Taylor's_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

<a href="Differentiation_rules" title="Differentiation rules">Rules and identities</a></th></tr><tr><td class="sidebar-content hlist">

<li><a href="Product_rule" title="Product rule">Product</a></li>

<li><a href="Chain_rule" title="Chain rule">Chain</a></li>

<li><a href="Power_rule" title="Power rule">Power</a></li>

<li><a href="Quotient_rule" title="Quotient rule">Quotient</a></li>

<li><a href="L'H%C3%B4pital's_rule" title="L'Hôpital's rule">L'Hôpital's rule</a></li>

<li><a href="Inverse_function_rule" title="Inverse function rule">Inverse</a></li>

<li><a href="General_Leibniz_rule" title="General Leibniz rule">General Leibniz</a></li>

<li><a href="Fa%C3%A0_di_Bruno's_formula" title="Faà di Bruno's formula">Faà di Bruno's formula</a></li>

<li><a href="Reynolds_transport_theorem" title="Reynolds transport theorem">Reynolds</a></li></ul></td>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="Integral" title="Integral">Integral</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">

<ul><li><a href="Lists_of_integrals" title="Lists of integrals">Lists of integrals</a></li>

<li><a href="Integral_transform" title="Integral transform">Integral transform</a></li>

<li><a href="Leibniz_integral_rule" title="Leibniz integral rule">Leibniz integral rule</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Definitions</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Antiderivative" title="Antiderivative">Antiderivative</a></li>

<li><a href="Integral" title="Integral">Integral</a> (<a href="Improper_integral" title="Improper integral">improper</a>)</li>

<li><a href="Riemann_integral" title="Riemann integral">Riemann integral</a></li>

<li><a href="Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li>

<li><a href="Contour_integration" title="Contour integration">Contour integration</a></li>

<li><a href="Integral_of_inverse_functions" title="Integral of inverse functions">Integral of inverse functions</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Integration by</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Integration_by_parts" title="Integration by parts">Parts</a></li>

<li><a href="Disc_integration" title="Disc integration">Discs</a></li>

<li><a href="Shell_integration" title="Shell integration">Cylindrical shells</a></li>

<li><a href="Integration_by_substitution" title="Integration by substitution">Substitution</a> (<a href="Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a>, <a href="Tangent_half-angle_substitution" title="Tangent half-angle substitution">tangent half-angle</a>, <a href="Euler_substitution" title="Euler substitution">Euler</a>)</li>

<li><a href="Integration_using_Euler's_formula" title="Integration using Euler's formula">Euler's formula</a></li>

<li><a href="Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions</a> (<a href="Heaviside_cover-up_method" title="Heaviside cover-up method">Heaviside's method</a>)</li>

<li><a href="Order_of_integration_(calculus)" title="Order of integration (calculus)">Changing order</a></li>

<li><a href="Integration_by_reduction_formulae" title="Integration by reduction formulae">Reduction formulae</a></li>

<li><a href="Leibniz_integral_rule#Evaluating_definite_integrals" title="Leibniz integral rule">Differentiating under the integral sign</a></li>

<li><a href="Risch_algorithm" title="Risch algorithm">Risch algorithm</a></li></ul></td>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="Series_(mathematics)" title="Series (mathematics)">Series</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">

<ul><li><a href="Geometric_series" title="Geometric series">Geometric</a> (<a href="Arithmetico%E2%80%93geometric_sequence" class="mw-redirect" title="Arithmetico–geometric sequence">arithmetico-geometric</a>)</li>

<li><a href="Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li>

<li><a href="Alternating_series" title="Alternating series">Alternating</a></li>

<li><a href="Power_series" title="Power series">Power</a></li>

<li><a href="Binomial_series" title="Binomial series">Binomial</a></li>

<li><a href="Taylor_series" title="Taylor series">Taylor</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

<a href="Convergence_tests" title="Convergence tests">Convergence tests</a></th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Term_test" class="mw-redirect" title="Term test">Summand limit (term test)</a></li>

<li><a href="Ratio_test" title="Ratio test">Ratio</a></li>

<li><a href="Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li>

<li><a href="Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li>

<li> <a href="Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li>

<li><a href="Alternating_series_test" title="Alternating series test">Alternating series</a></li>

<li><a href="Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li>

<li><a href="Dirichlet's_test" title="Dirichlet's test">Dirichlet</a></li>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="Vector_calculus" title="Vector calculus">Vector</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">

<ul><li><a href="Gradient" title="Gradient">Gradient</a></li>

<li><a href="Divergence" title="Divergence">Divergence</a></li>

<li><a href="Laplace_operator" title="Laplace operator">Laplacian</a></li>

<li><a href="Directional_derivative" title="Directional derivative">Directional derivative</a></li>

<li><a href="Vector_calculus_identities" title="Vector calculus identities">Identities</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Theorems</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Gradient_theorem" title="Gradient theorem">Gradient</a></li>

<li><a href="Green's_theorem" title="Green's theorem">Green's</a></li>

<li><a href="Stokes'_theorem" title="Stokes' theorem">Stokes'</a></li>

<li><a href="Divergence_theorem" title="Divergence theorem">Divergence</a></li>

<li><a href="Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li>

<li><a href="Helmholtz_decomposition" title="Helmholtz decomposition">Helmholtz decomposition</a></li></ul></td>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="Multivariable_calculus" title="Multivariable calculus">Multivariable</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading">

Formalisms</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Matrix_calculus" title="Matrix calculus">Matrix</a></li>

<li><a href="Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor</a></li>

<li><a href="Exterior_derivative" title="Exterior derivative">Exterior</a></li>

<li><a href="Geometric_calculus" title="Geometric calculus">Geometric</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Definitions</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Partial_derivative" title="Partial derivative">Partial derivative</a></li>

<li><a href="Multiple_integral" title="Multiple integral">Multiple integral</a></li>

<li><a href="Line_integral" title="Line integral">Line integral</a></li>

<li><a href="Surface_integral" title="Surface integral">Surface integral</a></li>

<li><a href="Volume_integral" title="Volume integral">Volume integral</a></li>

<li><a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a></li>

<li><a href="Hessian_matrix" title="Hessian matrix">Hessian</a></li></ul></td>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Advanced</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">

<ul><li><a href="Calculus_on_Euclidean_space" title="Calculus on Euclidean space">Calculus on Euclidean space</a></li>

<li><a href="Generalized_function" title="Generalized function">Generalized functions</a></li>

<li><a href="Limit_of_distributions" title="Limit of distributions">Limit of distributions</a></li></ul></td>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Specialized</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;">

<ul><li><a href="Fractional_calculus" title="Fractional calculus">Fractional</a></li>

<li><a href="Malliavin_calculus" title="Malliavin calculus">Malliavin</a></li>

<li><a href="Stochastic_calculus" title="Stochastic calculus">Stochastic</a></li>

<li><a href="Calculus_of_variations" title="Calculus of variations">Variations</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Miscellanea</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;">

<ul><li><a href="Precalculus" title="Precalculus">Precalculus</a></li>

<li><a href="History_of_calculus" title="History of calculus">History</a></li>

<li><a href="Glossary_of_calculus" title="Glossary of calculus">Glossary</a></li>

<li><a href="List_of_calculus_topics" title="List of calculus topics">List of topics</a></li>

<li><a href="Integration_Bee" title="Integration Bee">Integration Bee</a></li>

<li><a href="Mathematical_analysis" title="Mathematical analysis">Mathematical analysis</a></li>

<li><a href="Nonstandard_analysis" title="Nonstandard analysis">Nonstandard analysis</a></li></ul></div></div></td>

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In <a href="Mathematics" title="Mathematics">mathematics</a>, a continuous function is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> such that a small variation of the <a href="Argument_of_a_function" title="Argument of a function">argument</a> induces a small variation of the <a href="Value_(mathematics)" title="Value (mathematics)">value</a> of the function. This implies there are no abrupt changes in value, known as <a href="Classification_of_discontinuities" title="Classification of discontinuities">discontinuities</a>. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is not continuous. Until the 19th century, mathematicians largely relied on <a href="Intuition" title="Intuition">intuitive</a> notions of continuity and considered only continuous functions. The <a href="(%CE%B5%2C_%CE%B4)-definition_of_limit" class="mw-redirect" title="(ε, δ)-definition of limit">epsilon–delta definition of a limit</a> was introduced to formalize the definition of continuity.

Continuity is one of the core concepts of <a href="Calculus" title="Calculus">calculus</a> and <a href="Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, where arguments and values of functions are <a href="Real_number" title="Real number">real</a> and <a href="Complex_number" title="Complex number">complex</a> numbers. The concept has been generalized to functions <a href="#Continuous_functions_between_metric_spaces">between metric spaces</a> and <a href="#Continuous_functions_between_topological_spaces">between topological spaces</a>. The latter are the most general continuous functions, and their definition is the basis of <a href="Topology" title="Topology">topology</a>.

A stronger form of continuity is <a href="Uniform_continuity" title="Uniform continuity">uniform continuity</a>. In <a href="Order_theory" title="Order theory">order theory</a>, especially in <a href="Domain_theory" title="Domain theory">domain theory</a>, a related concept of continuity is <a href="Scott_continuity" title="Scott continuity">Scott continuity</a>.

As an example, the function H(t) denoting the height of a growing flower at time t would be considered continuous. In contrast, the function M(t) denoting the amount of money in a bank account at time t would be considered discontinuous since it "jumps" at each point in time when money is deposited or withdrawn.

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

A form of the <a href="Limit_of_a_function#(ε,_δ)-definition_of_limit" title="Limit of a function">epsilon–delta definition of continuity</a> was first given by <a href="Bernard_Bolzano" title="Bernard Bolzano">Bernard Bolzano</a> in 1817. <a href="Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a> defined continuity of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)}">

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<annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation>

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</math><img src="./2311a6a75c54b0ea085a381ba472c31d59321514.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}" loading="lazy"> as follows: an infinitely small increment <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> of the independent variable x always produces an infinitely small change <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x+\alpha )-f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x+\alpha )-f(x)}</annotation>

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</math><img src="./8cdc7a8529de4a0492b62de55ca5c0bc15ca2651.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.004ex; height:2.843ex;" alt="{\displaystyle f(x+\alpha )-f(x)}" loading="lazy"> of the dependent variable y (see e.g. <a href="Cours_d'Analyse" class="mw-redirect" title="Cours d'Analyse">Cours d'Analyse</a>, p. 34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today (see <a href="Microcontinuity" title="Microcontinuity">microcontinuity</a>). The formal definition and the distinction between pointwise continuity and <a href="Uniform_continuity" title="Uniform continuity">uniform continuity</a> were first given by Bolzano in the 1830s, but the work wasn't published until the 1930s. Like Bolzano,<a href="#cite_note-1">[1]</a> <a href="Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a><a href="#cite_note-2">[2]</a> denied continuity of a function at a point c unless it was defined at and on both sides of c, but <a href="%C3%89douard_Goursat" title="Édouard Goursat">Édouard Goursat</a><a href="#cite_note-3">[3]</a> allowed the function to be defined only at and on one side of c, and <a href="Camille_Jordan" title="Camille Jordan">Camille Jordan</a><a href="#cite_note-4">[4]</a> allowed it even if the function was defined only at c. All three of those nonequivalent definitions of pointwise continuity are still in use.<a href="#cite_note-5">[5]</a> <a href="Eduard_Heine" title="Eduard Heine">Eduard Heine</a> provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by <a href="Peter_Gustav_Lejeune_Dirichlet" title="Peter Gustav Lejeune Dirichlet">Peter Gustav Lejeune Dirichlet</a> in 1854.<a href="#cite_note-6">[6]</a>

<div class="mw-heading mw-heading2"><h2 id="Real_functions">Real functions</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Definition">Definition</h3></div>

A <a href="Real_function" class="mw-redirect" title="Real function">real function</a> that is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> from <a href="Real_number" title="Real number">real numbers</a> to real numbers can be represented by a <a href="Graph_of_a_function" title="Graph of a function">graph</a> in the <a href="Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian plane</a>; such a function is continuous if, roughly speaking, the graph is a single unbroken <a href="Curve" title="Curve">curve</a> whose <a href="Domain_of_a_function" title="Domain of a function">domain</a> is the entire real line. A more mathematically rigorous definition is given below.<a href="#cite_note-8">[8]</a>

Continuity of real functions is usually defined in terms of <a href="Limit_(mathematics)" title="Limit (mathematics)">limits</a>. A function f with variable x is continuous at the <a href="Real_number" title="Real number">real number</a> c, if the limit of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x),}">

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<annotation encoding="application/x-tex">{\displaystyle f(x),}</annotation>

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</math><img src="./10535d1a7a971ffeeb216605cb846099fab2e653.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.064ex; height:2.843ex;" alt="{\displaystyle f(x),}" loading="lazy"> as x tends to c, is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(c).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(c).}</annotation>

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</math><img src="./4c253cbfe51ac86f97d50d64a343bf30e62063ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.742ex; height:2.843ex;" alt="{\displaystyle f(c).}" loading="lazy">

There are several different definitions of the (global) continuity of a function, which depend on the nature of its <a href="Domain_of_a_function" title="Domain of a function">domain</a>.

A function is continuous on an <a href="Open_interval" class="mw-redirect" title="Open interval">open interval</a> if the interval is contained in the function's domain and the function is continuous at every interval point. A function that is continuous on the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,+\infty )}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (-\infty ,+\infty )}</annotation>

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</math><img src="./e577bfa9ed1c0f83ed643206abae3cd2f234cf9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.107ex; height:2.843ex;" alt="{\displaystyle (-\infty ,+\infty )}" loading="lazy"> (the whole <a href="Real_line" class="mw-redirect" title="Real line">real line</a>) is often called simply a continuous function; one also says that such a function is continuous everywhere. For example, all <a href="Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial functions</a> are continuous everywhere.

A function is continuous on a <a href="Semi-open_interval" class="mw-redirect" title="Semi-open interval">semi-open</a> or a <a href="Closed_interval" class="mw-redirect" title="Closed interval">closed</a> interval; if the interval is contained in the domain of the function, the function is continuous at every interior point of the interval, and the value of the function at each endpoint that belongs to the interval is the limit of the values of the function when the variable tends to the endpoint from the interior of the interval. For example, the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\sqrt {x}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)={\sqrt {x}}}</annotation>

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</math><img src="./7f4850536e7a37db22aacbc552b03f195a3eceaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.782ex; height:3.009ex;" alt="{\displaystyle f(x)={\sqrt {x}}}" loading="lazy"> is continuous on its whole domain, which is the closed interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,+\infty ).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,+\infty ).}</annotation>

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</math><img src="./454e2acfd64f7f36372f601402e502125ddddf8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.527ex; height:2.843ex;" alt="{\displaystyle [0,+\infty ).}" loading="lazy">

Many commonly encountered functions are <a href="Partial_function" title="Partial function">partial functions</a> that have a domain formed by all real numbers, except some <a href="Isolated_point" title="Isolated point">isolated points</a>. Examples include the <a href="Reciprocal_function" class="mw-redirect" title="Reciprocal function">reciprocal function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x\mapsto {\frac {1}{x}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle x\mapsto {\frac {1}{x}}}</annotation>

</semantics>

</math><img src="./c4e512047f779008dd32f5b041baceb2c4dfa9a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.72ex; height:3.343ex;" alt="{\textstyle x\mapsto {\frac {1}{x}}}" loading="lazy"> and the <a href="Tangent_function" class="mw-redirect" title="Tangent function">tangent function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto \tan x.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\mapsto \tan x.}</annotation>

</semantics>

</math><img src="./27b4c4d0592ddd7c4163b6d448da41268b9d6ecd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.667ex; height:2.009ex;" alt="{\displaystyle x\mapsto \tan x.}" loading="lazy"> When they are continuous on their domain, one says, in some contexts, that they are continuous, although they are not continuous everywhere. In other contexts, mainly when one is interested in their behavior near the exceptional points, one says they are discontinuous.

A partial function is discontinuous at a point if the point belongs to the <a href="Topological_closure" class="mw-redirect" title="Topological closure">topological closure</a> of its domain, and either the point does not belong to the domain of the function or the function is not continuous at the point. For example, the functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x\mapsto {\frac {1}{x}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle x\mapsto {\frac {1}{x}}}</annotation>

</semantics>

</math><img src="./c4e512047f779008dd32f5b041baceb2c4dfa9a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.72ex; height:3.343ex;" alt="{\textstyle x\mapsto {\frac {1}{x}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x\mapsto \sin({\frac {1}{x}})}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle x\mapsto \sin({\frac {1}{x}})}</annotation>

</semantics>

</math><img src="./3570c0c792ba7fd90065ecc9291b52ee5d7ce127.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.385ex; height:3.343ex;" alt="{\textstyle x\mapsto \sin({\frac {1}{x}})}" loading="lazy"> are discontinuous at 0, and remain discontinuous whichever value is chosen for defining them at 0. A point where a function is discontinuous is called a discontinuity.

Using mathematical notation, several ways exist to define continuous functions in the three senses mentioned above.

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f:D\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f:D\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./ad6f25caa967d66ff907fb2faf36316ac557c589.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\textstyle f:D\to \mathbb {R} }" loading="lazy"> be a function whose <a href="Domain_of_a_function" title="Domain of a function">domain</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> is contained in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> of real numbers.

Some (but not all) possibilities for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> are:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> is the whole <a href="Real_line" class="mw-redirect" title="Real line">real line</a>; that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=\mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D=\mathbb {R} }</annotation>

</semantics>

</math><img src="./cce378810ddc483757baa7eee22efcaf6e271e64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.701ex; height:2.176ex;" alt="{\displaystyle D=\mathbb {R} }" loading="lazy"></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> is a <a href="Closed_interval" class="mw-redirect" title="Closed interval">closed interval</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=[a,b]=\{x\in \mathbb {R} \mid a\leq x\leq b\},}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D=[a,b]=\{x\in \mathbb {R} \mid a\leq x\leq b\},}</annotation>

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</math><img src="./dc3f2152a27a96fd31caeae439cba949a53a0b67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.187ex; height:2.843ex;" alt="{\displaystyle D=[a,b]=\{x\in \mathbb {R} \mid a\leq x\leq b\},}" loading="lazy"> where a and b are real numbers</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> is an <a href="Open_interval" class="mw-redirect" title="Open interval">open interval</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=(a,b)=\{x\in \mathbb {R} \mid a<x<b\},}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D=(a,b)=\{x\in \mathbb {R} \mid a<x<b\},}</annotation>

</semantics>

</math><img src="./dd4b2e4ddd01af2481580fce8daee5b9c6dc2c14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.703ex; height:2.843ex;" alt="{\displaystyle D=(a,b)=\{x\in \mathbb {R} \mid a<x<b\},}" loading="lazy"> where a and b are real numbers</li></ul>

In the case of an open interval, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> do not belong to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy">, and the values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(a)}</annotation>

</semantics>

</math><img src="./368cb4b81ba5754d7a354a4ce49c2f1084bdaace.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.318ex; height:2.843ex;" alt="{\displaystyle f(a)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(b)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(b)}</annotation>

</semantics>

</math><img src="./adbef9aaf500c30a3f107e5d7efb7a8627d3bba9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.085ex; height:2.843ex;" alt="{\displaystyle f(b)}" loading="lazy"> are not defined, and if they are, they do not matter for continuity on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Definition_in_terms_of_limits_of_functions">Definition in terms of limits of functions</h4></div>

The function f is continuous at some point c of its domain if the <a href="Limit_of_a_function" title="Limit of a function">limit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x),}</annotation>

</semantics>

</math><img src="./10535d1a7a971ffeeb216605cb846099fab2e653.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.064ex; height:2.843ex;" alt="{\displaystyle f(x),}" loading="lazy"> as x approaches c through the domain of f, exists and is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(c).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(c).}</annotation>

</semantics>

</math><img src="./4c253cbfe51ac86f97d50d64a343bf30e62063ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.742ex; height:2.843ex;" alt="{\displaystyle f(c).}" loading="lazy"><a href="#cite_note-9">[9]</a> In mathematical notation, this is written as

</mrow>

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{x\to c}{f(x)}=f(c).}</annotation>

</semantics>

</math>

In detail this means three conditions: first, f has to be defined at c (guaranteed by the requirement that c is in the domain of f). Second, the limit of that equation has to exist. Third, the value of this limit must equal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(c).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(c).}</annotation>

</semantics>

</math><img src="./4c253cbfe51ac86f97d50d64a343bf30e62063ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.742ex; height:2.843ex;" alt="{\displaystyle f(c).}" loading="lazy">

(Here, we have assumed that the domain of f does not have any <a href="Isolated_point" title="Isolated point">isolated points</a>.)

<div class="mw-heading mw-heading4"><h4 id="Definition_in_terms_of_neighborhoods">Definition in terms of neighborhoods</h4></div>

A <a href="Neighborhood_(mathematics)" class="mw-redirect" title="Neighborhood (mathematics)">neighborhood</a> of a point c is a set that contains, at least, all points within some fixed distance of c. Intuitively, a function is continuous at a point c if the range of f over the neighborhood of c shrinks to a single point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(c)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(c)}</annotation>

</semantics>

</math><img src="./97d4cbf97e497bd1b20228d6a1c12f5f34b87213.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.095ex; height:2.843ex;" alt="{\displaystyle f(c)}" loading="lazy"> as the width of the neighborhood around c shrinks to zero. More precisely, a function f is continuous at a point c of its domain if, for any neighborhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{1}(f(c))}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N_{1}(f(c))}</annotation>

</semantics>

</math><img src="./0a0caf40bdbfade8d474850b998a0d4e6a27dc68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.824ex; height:2.843ex;" alt="{\displaystyle N_{1}(f(c))}" loading="lazy"> there is a neighborhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{2}(c)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N_{2}(c)}</annotation>

</semantics>

</math><img src="./18e9c410ddb370a1c355e386ec67de85190f2153.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.737ex; height:2.843ex;" alt="{\displaystyle N_{2}(c)}" loading="lazy"> in its domain such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\in N_{1}(f(c))}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)\in N_{1}(f(c))}</annotation>

</semantics>

</math><img src="./8a64d5ab6d49d8d160869f7b07d34eb89e35bd0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.083ex; height:2.843ex;" alt="{\displaystyle f(x)\in N_{1}(f(c))}" loading="lazy"> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in N_{2}(c).}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in N_{2}(c).}</annotation>

</semantics>

</math><img src="./9a0ea5a03bd110a299ef8bb0767ca65b83a45552.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.554ex; height:2.843ex;" alt="{\displaystyle x\in N_{2}(c).}" loading="lazy">

As neighborhoods are defined in any <a href="Topological_space" title="Topological space">topological space</a>, this definition of a continuous function applies not only for real functions but also when the domain and the <a href="Codomain" title="Codomain">codomain</a> are <a href="Topological_space" title="Topological space">topological spaces</a> and is thus the most general definition. It follows that a function is automatically continuous at every <a href="Isolated_point" title="Isolated point">isolated point</a> of its domain. For example, every real-valued function on the integers is continuous.

<div class="mw-heading mw-heading4"><h4 id="Definition_in_terms_of_limits_of_sequences">Definition in terms of limits of sequences</h4></div>

One can instead require that for any <a href="Sequence_(mathematics)" class="mw-redirect" title="Sequence (mathematics)">sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{n})_{n\in \mathbb {N} }}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{n})_{n\in \mathbb {N} }}</annotation>

</semantics>

</math><img src="./d9ebf5fbc73781898c4395888facf8a8ef83a7cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.859ex; height:2.843ex;" alt="{\displaystyle (x_{n})_{n\in \mathbb {N} }}" loading="lazy"> of points in the domain which <a href="Convergent_sequence" class="mw-redirect" title="Convergent sequence">converges</a> to c, the corresponding sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(f(x_{n})\right)_{n\in \mathbb {N} }}">

<msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(f(x_{n})\right)_{n\in \mathbb {N} }}</annotation>

</semantics>

</math><img src="./86088244989913be2b2555cb16a134785b5e2f90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.947ex; height:3.009ex;" alt="{\displaystyle \left(f(x_{n})\right)_{n\in \mathbb {N} }}" loading="lazy"> converges to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(c).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(c).}</annotation>

</semantics>

</math><img src="./4c253cbfe51ac86f97d50d64a343bf30e62063ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.742ex; height:2.843ex;" alt="{\displaystyle f(c).}" loading="lazy"> In mathematical notation, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall (x_{n})_{n\in \mathbb {N} }\subset D:\lim _{n\to \infty }x_{n}=c\Rightarrow \lim _{n\to \infty }f(x_{n})=f(c)\,.}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \forall (x_{n})_{n\in \mathbb {N} }\subset D:\lim _{n\to \infty }x_{n}=c\Rightarrow \lim _{n\to \infty }f(x_{n})=f(c)\,.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading4"><h4 id="Weierstrass_and_Jordan_definitions_(epsilon–delta)_of_continuous_functions">Weierstrass and Jordan definitions (epsilon–delta) of continuous functions</h4></div>

Explicitly including the definition of the limit of a function, we obtain a self-contained definition: Given a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:D\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:D\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./b43114d77160d4967e4b729e8eac6e4555f15300.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f:D\to \mathbb {R} }" loading="lazy"> as above and an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> of the domain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is said to be continuous at the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> when the following holds: For any positive real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon >0,}</annotation>

</semantics>

</math><img src="./4d780d8dff4b26013c7d5d0efbc1acb92b60645b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.991ex; height:2.509ex;" alt="{\displaystyle \varepsilon >0,}" loading="lazy"> however small, there exists some positive real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta >0}</annotation>

</semantics>

</math><img src="./595d5cea06fdcaf2642caf549eda2cfc537958a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta >0}" loading="lazy"> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> in the domain of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}-\delta <x<x_{0}+\delta ,}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}-\delta <x<x_{0}+\delta ,}</annotation>

</semantics>

</math><img src="./56eb6512d1de61a0374678fe5845f30826643e2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.719ex; height:2.676ex;" alt="{\displaystyle x_{0}-\delta <x<x_{0}+\delta ,}" loading="lazy"> the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> satisfies

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\left(x_{0}\right)-\varepsilon <f(x)<f(x_{0})+\varepsilon .}</annotation>

</semantics>

</math>

Alternatively written, continuity of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:D\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:D\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./b43114d77160d4967e4b729e8eac6e4555f15300.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f:D\to \mathbb {R} }" loading="lazy"> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in D}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}\in D}</annotation>

</semantics>

</math><img src="./b107b130543b591ac3037f4bc152da314ec51586.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.149ex; height:2.509ex;" alt="{\displaystyle x_{0}\in D}" loading="lazy"> means that for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon >0,}</annotation>

</semantics>

</math><img src="./4d780d8dff4b26013c7d5d0efbc1acb92b60645b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.991ex; height:2.509ex;" alt="{\displaystyle \varepsilon >0,}" loading="lazy"> there exists a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta >0}</annotation>

</semantics>

</math><img src="./595d5cea06fdcaf2642caf549eda2cfc537958a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta >0}" loading="lazy"> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in D}</annotation>

</semantics>

</math><img src="./aaf3e2c3607ccf1c1d7ee6620b44ef9d9e2e1f6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.095ex; height:2.176ex;" alt="{\displaystyle x\in D}" loading="lazy">:

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mtext> implies </mtext>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left|x-x_{0}\right|<\delta ~~{\text{ implies }}~~|f(x)-f(x_{0})|<\varepsilon .}</annotation>

</semantics>

</math>

More intuitively, we can say that if we want to get all the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> values to stay in some small <a href="Topological_neighborhood" class="mw-redirect" title="Topological neighborhood">neighborhood</a> around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(x_{0}\right),}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\left(x_{0}\right),}</annotation>

</semantics>

</math><img src="./481e50f14b16535e7c3f3bb1001bd9e6dab77246.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.893ex; height:2.843ex;" alt="{\displaystyle f\left(x_{0}\right),}" loading="lazy"> we need to choose a small enough neighborhood for the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> values around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}.}</annotation>

</semantics>

</math><img src="./7ec756993d89cc1bd74f84040c07f5e11f0a8102.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.031ex; height:2.009ex;" alt="{\displaystyle x_{0}.}" loading="lazy"> If we can do that no matter how small the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{0})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x_{0})}</annotation>

</semantics>

</math><img src="./27cf1dbaefc6038a22779fb2943aff758a592a3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.472ex; height:2.843ex;" alt="{\displaystyle f(x_{0})}" loading="lazy"> neighborhood is, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is continuous at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}.}</annotation>

</semantics>

</math><img src="./7ec756993d89cc1bd74f84040c07f5e11f0a8102.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.031ex; height:2.009ex;" alt="{\displaystyle x_{0}.}" loading="lazy">

In modern terms, this is generalized by the definition of continuity of a function with respect to a <a href="Basis_(topology)" class="mw-redirect" title="Basis (topology)">basis for the topology</a>, here the <a href="Metric_topology" class="mw-redirect" title="Metric topology">metric topology</a>.

Weierstrass had required that the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}-\delta <x<x_{0}+\delta }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}-\delta <x<x_{0}+\delta }</annotation>

</semantics>

</math><img src="./09d4ee2f191dcd60dde2cde40f56ecdbb9452460.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.073ex; height:2.676ex;" alt="{\displaystyle x_{0}-\delta <x<x_{0}+\delta }" loading="lazy"> be entirely within the domain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy">, but Jordan removed that restriction.

<div class="mw-heading mw-heading4"><h4 id="Definition_in_terms_of_control_of_the_remainder">Definition in terms of control of the remainder</h4></div>

In proofs and numerical analysis, we often need to know how fast limits are converging, or in other words, control of the remainder. We can formalize this to a definition of continuity.

A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C:[0,\infty )\to [0,\infty ]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C:[0,\infty )\to [0,\infty ]}</annotation>

</semantics>

</math><img src="./96dbec0587155c14584a3c95ebf835b19a3fa57b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.203ex; height:2.843ex;" alt="{\displaystyle C:[0,\infty )\to [0,\infty ]}" loading="lazy"> is called a control function if

<ul><li>C is non-decreasing</li>

<li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \inf _{\delta >0}C(\delta )=0}">

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \inf _{\delta >0}C(\delta )=0}</annotation>

</semantics>

</math><img src="./dc5fd21047e64de2fd6d90a3538f7a1359d516d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.114ex; height:4.176ex;" alt="{\displaystyle \inf _{\delta >0}C(\delta )=0}" loading="lazy"></li></ul>

A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:D\to R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:D\to R}</annotation>

</semantics>

</math><img src="./66c6aa28d2db06a0437aaac9cef5c7a011b513c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.518ex; height:2.509ex;" alt="{\displaystyle f:D\to R}" loading="lazy"> is C-continuous at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> if there exists such a neighbourhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle N(x_{0})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle N(x_{0})}</annotation>

</semantics>

</math><img src="./172a13dfbe4d6f6ffb1a113c851722ce85850ee2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.257ex; height:2.843ex;" alt="{\textstyle N(x_{0})}" loading="lazy"> that

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |f(x)-f(x_{0})|\leq C\left(\left|x-x_{0}\right|\right){\text{ for all }}x\in D\cap N(x_{0})}</annotation>

</semantics>

</math>

A function is continuous in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> if it is C-continuous for some control function C.

This approach leads naturally to refining the notion of continuity by restricting the set of admissible control functions. For a given set of control functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> a function is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy">-continuous if it is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">-continuous for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\in {\mathcal {C}}.}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C\in {\mathcal {C}}.}</annotation>

</semantics>

</math><img src="./10036bb311876f4dfc3ca6832e7ace30b9c06fc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.493ex; height:2.176ex;" alt="{\displaystyle C\in {\mathcal {C}}.}" loading="lazy"> For example, the <a href="Lipschitz_continuity" title="Lipschitz continuity">Lipschitz</a>, the <a href="H%C3%B6lder_continuous_function" class="mw-redirect" title="Hölder continuous function">Hölder continuous functions</a> of exponent α and the <a href="Uniformly_continuous_function" class="mw-redirect" title="Uniformly continuous function">uniformly continuous functions</a> below are defined by the set of control functions

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}_{\mathrm {Lipschitz} }=\{C:C(\delta )=K|\delta |,\ K>0\}}</annotation>

</semantics>

</math>

<msub>

</mrow>

<mtext>Hölder</mtext>

</mrow>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}_{{\text{Hölder}}-\alpha }=\{C:C(\delta )=K|\delta |^{\alpha },\ K>0\}}</annotation>

</semantics>

</math>

<msub>

</mrow>

<mtext>uniform cont.</mtext>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}_{\text{uniform cont.}}=\{C:C(0)=0\}}</annotation>

</semantics>

</math>

respectively.

<div class="mw-heading mw-heading4"><h4 id="Definition_using_oscillation">Definition using oscillation</h4></div>

Continuity can also be defined in terms of <a href="Oscillation_(mathematics)" title="Oscillation (mathematics)">oscillation</a>: a function f is continuous at a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> if and only if its oscillation at that point is zero;<a href="#cite_note-10">[10]</a> in symbols, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{f}(x_{0})=0.}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{f}(x_{0})=0.}</annotation>

</semantics>

</math><img src="./1c268cf8439c21dca12057806d013aa2e34a304a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.683ex; height:3.009ex;" alt="{\displaystyle \omega _{f}(x_{0})=0.}" loading="lazy"> A benefit of this definition is that it quantifies discontinuity: the oscillation gives how much the function is discontinuous at a point.

This definition is helpful in <a href="Descriptive_set_theory" title="Descriptive set theory">descriptive set theory</a> to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

</semantics>

</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"> (hence a <a href="G-delta_set" class="mw-redirect" title="G-delta set"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\delta }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{\delta }}</annotation>

</semantics>

</math><img src="./7b42b79015590af928aa28bda9514373aeac0e1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.8ex; height:2.509ex;" alt="{\displaystyle G_{\delta }}" loading="lazy"> set</a>) – and gives a rapid proof of one direction of the <a href="Lebesgue_integrability_condition" class="mw-redirect" title="Lebesgue integrability condition">Lebesgue integrability condition</a>.<a href="#cite_note-11">[11]</a>

The oscillation is equivalent to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon -\delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon -\delta }</annotation>

</semantics>

</math><img src="./bdcdab850a3df0b5c13909e4b058fb35220f019d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.973ex; height:2.509ex;" alt="{\displaystyle \varepsilon -\delta }" loading="lazy"> definition by a simple re-arrangement and by using a limit (<a href="Lim_sup" class="mw-redirect" title="Lim sup">lim sup</a>, <a href="Lim_inf" class="mw-redirect" title="Lim inf">lim inf</a>) to define oscillation: if (at a given point) for a given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}}</annotation>

</semantics>

</math><img src="./acb0a8377db20e42274444cb181d51b5532b5844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{0}}" loading="lazy"> there is no <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta }</annotation>

</semantics>

</math><img src="./c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"> that satisfies the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon -\delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon -\delta }</annotation>

</semantics>

</math><img src="./bdcdab850a3df0b5c13909e4b058fb35220f019d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.973ex; height:2.509ex;" alt="{\displaystyle \varepsilon -\delta }" loading="lazy"> definition, then the oscillation is at least <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{0},}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0},}</annotation>

</semantics>

</math><img src="./c6feb7a4465d20441c2c1e361935a4d5035ff2c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.785ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{0},}" loading="lazy"> and conversely if for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

</semantics>

</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"> there is a desired <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta ,}</annotation>

</semantics>

</math><img src="./019f26293a6ac6d9e1521a005825cc3716cf12ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.695ex; height:2.676ex;" alt="{\displaystyle \delta ,}" loading="lazy"> the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a <a href="Metric_space" title="Metric space">metric space</a>.

<div class="mw-heading mw-heading4"><h4 id="Definition_using_the_hyperreals">Definition using the hyperreals</h4></div>

<a href="Cauchy" class="mw-redirect" title="Cauchy">Cauchy</a> defined the continuity of a function in the following intuitive terms: an <a href="Infinitesimal" title="Infinitesimal">infinitesimal</a> change in the independent variable corresponds to an infinitesimal change of the dependent variable (see Cours d'analyse, page 34). <a href="Non-standard_analysis" class="mw-redirect" title="Non-standard analysis">Non-standard analysis</a> is a way of making this mathematically rigorous. The real line is augmented by adding infinite and infinitesimal numbers to form the <a href="Hyperreal_numbers" class="mw-redirect" title="Hyperreal numbers">hyperreal numbers</a>. In nonstandard analysis, continuity can be defined as follows.

/* start https://en.wikipedia.org/ */

.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}

/* end https://en.wikipedia.org/ */

</style><div class="block-indent" style="padding-left: 1.5em;">A real-valued function f is continuous at x if its natural extension to the hyperreals has the property that for all infinitesimal dx, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x+dx)-f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x+dx)-f(x)}</annotation>

</semantics>

</math><img src="./0493531da57357fa09e35ce1f57d89fe541a7085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.061ex; height:2.843ex;" alt="{\displaystyle f(x+dx)-f(x)}" loading="lazy"> is infinitesimal<a href="#cite_note-12">[12]</a></div>

(see <a href="Microcontinuity" title="Microcontinuity">microcontinuity</a>). In other words, an infinitesimal increment of the independent variable always produces an infinitesimal change of the dependent variable, giving a modern expression to <a href="Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a>'s definition of continuity.

<div class="mw-heading mw-heading3"><h3 id="Rules_for_continuity">Rules for continuity</h3></div>

Proving the continuity of a function by a direct application of the definition is generaly a noneasy task. Fortunately, in practice, most functions are built from simpler functions, and their continuity can be deduced immediately from the way they are defined, by applying the following rules:

<ul><li>Every <a href="Constant_function" title="Constant function">constant function</a> is continuous</li>

<li>The <a href="Identity_function" title="Identity function">identity function</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=x}</annotation>

</semantics>

</math><img src="./f690285952308aa49e3c6aac892df31cad6d1b06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.846ex; height:2.843ex;" alt="{\displaystyle f(x)=x}" loading="lazy">⁠ is continuous</li>

<li>Addition and multiplication: If the functions ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy">⁠ are continuous on their respective domains ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{f}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{f}}</annotation>

</semantics>

</math><img src="./13b8420d715315c8e8b57f178a1b3a8512bd0959.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.061ex; height:2.843ex;" alt="{\displaystyle D_{f}}" loading="lazy">⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{g}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{g}}</annotation>

</semantics>

</math><img src="./6f5626aa4260303f293c7ebaa4786e38440557c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.946ex; height:2.843ex;" alt="{\displaystyle D_{g}}" loading="lazy">⁠, then their sum ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f+g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f+g}</annotation>

</semantics>

</math><img src="./4d94a24abd865f6f9fd67a7df7e531cae1c769b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.235ex; height:2.509ex;" alt="{\displaystyle f+g}" loading="lazy">⁠ and their product ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\cdot g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\cdot g}</annotation>

</semantics>

</math><img src="./95c09ad821122bb3fd64953313451b9267f75389.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.074ex; height:2.509ex;" alt="{\displaystyle f\cdot g}" loading="lazy">⁠ are continuous on the <a href="Set_intersection" class="mw-redirect" title="Set intersection">intersection</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{f}\cap D_{g}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{f}\cap D_{g}}</annotation>

</semantics>

</math><img src="./806d0c94c408ddb8f1ebbf270c4a2a31d357d2e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.589ex; height:2.843ex;" alt="{\displaystyle D_{f}\cap D_{g}}" loading="lazy">⁠, where ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f+g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f+g}</annotation>

</semantics>

</math><img src="./4d94a24abd865f6f9fd67a7df7e531cae1c769b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.235ex; height:2.509ex;" alt="{\displaystyle f+g}" loading="lazy">⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle fg}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle fg}</annotation>

</semantics>

</math><img src="./06bac4638bb56f14688118ce88c188c7a021eb29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.395ex; height:2.509ex;" alt="{\displaystyle fg}" loading="lazy">⁠ are defined by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f+g)(x)=f(x)+g(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f+g)(x)=f(x)+g(x)}</annotation>

</semantics>

</math><img src="./cf80cb50218eac1e40d4a0908bd039db3bd0863c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.795ex; height:2.843ex;" alt="{\displaystyle (f+g)(x)=f(x)+g(x)}" loading="lazy">⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f\cdot g)(x)=f(x)\cdot g(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f\cdot g)(x)=f(x)\cdot g(x)}</annotation>

</semantics>

</math><img src="./11b7dd97937ffc4103e823272727b6a4d9a36e44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.472ex; height:2.843ex;" alt="{\displaystyle (f\cdot g)(x)=f(x)\cdot g(x)}" loading="lazy">⁠.</li>

<li><a href="Multiplicative_inverse" title="Multiplicative inverse">Reciprocal</a>: If the function ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">⁠ is continuous on the domain ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{f}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{f}}</annotation>

</semantics>

</math><img src="./13b8420d715315c8e8b57f178a1b3a8512bd0959.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.061ex; height:2.843ex;" alt="{\displaystyle D_{f}}" loading="lazy">⁠, then its reciprocal ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{f}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{f}}}</annotation>

</semantics>

</math><img src="./14e22b329663dcafd279fbbf565b682bcfc22fb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:1.74ex; height:4.009ex;" alt="{\displaystyle {\tfrac {1}{f}}}" loading="lazy">⁠, defined by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\tfrac {1}{f}})(x)={\tfrac {1}{f(x)}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\tfrac {1}{f}})(x)={\tfrac {1}{f(x)}}}</annotation>

</semantics>

</math><img src="./33661cdd211604127afd9a5dc3113b231b712bbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.747ex; height:4.176ex;" alt="{\displaystyle ({\tfrac {1}{f}})(x)={\tfrac {1}{f(x)}}}" loading="lazy">⁠ is continuous on the domain ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{f}\setminus f^{-1}(0)}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{f}\setminus f^{-1}(0)}</annotation>

</semantics>

</math><img src="./8f2dd3a736c34308147a42aff623b117ff396b3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.88ex; height:3.343ex;" alt="{\displaystyle D_{f}\setminus f^{-1}(0)}" loading="lazy">⁠, that is, the domain ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{f}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{f}}</annotation>

</semantics>

</math><img src="./13b8420d715315c8e8b57f178a1b3a8512bd0959.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.061ex; height:2.843ex;" alt="{\displaystyle D_{f}}" loading="lazy">⁠ from which the points ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">⁠ such that ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=0}</annotation>

</semantics>

</math><img src="./cf85883d74b75fe35ca8d3f2b44802df078e4fa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="{\displaystyle f(x)=0}" loading="lazy">⁠ are removed.</li>

<li><a href="Function_composition" title="Function composition">Function composition</a>: If the functions ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy">⁠ are continuous on their respective domains ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{f}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{f}}</annotation>

</semantics>

</math><img src="./13b8420d715315c8e8b57f178a1b3a8512bd0959.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.061ex; height:2.843ex;" alt="{\displaystyle D_{f}}" loading="lazy">⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{g}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{g}}</annotation>

</semantics>

</math><img src="./6f5626aa4260303f293c7ebaa4786e38440557c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.946ex; height:2.843ex;" alt="{\displaystyle D_{g}}" loading="lazy">⁠, then the composition ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\circ f}</annotation>

</semantics>

</math><img src="./10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle g\circ f}" loading="lazy">⁠ defined by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {1}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {1}}</annotation>

</semantics>

</math><img src="./9c209c2b7f722d78521ecf455dd33a01bd8e48e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {1}}" loading="lazy">⁠ is continuous on ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{f}\cap f^{-1}(D_{g})}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{f}\cap f^{-1}(D_{g})}</annotation>

</semantics>

</math><img src="./cce1c4e1ffd27cd727b81e2452a4087561f1bc9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.051ex; height:3.343ex;" alt="{\displaystyle D_{f}\cap f^{-1}(D_{g})}" loading="lazy">⁠, that the part of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{f}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{f}}</annotation>

</semantics>

</math><img src="./13b8420d715315c8e8b57f178a1b3a8512bd0959.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.061ex; height:2.843ex;" alt="{\displaystyle D_{f}}" loading="lazy">⁠ that is mapped by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">⁠ inside ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{g}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{g}}</annotation>

</semantics>

</math><img src="./6f5626aa4260303f293c7ebaa4786e38440557c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.946ex; height:2.843ex;" alt="{\displaystyle D_{g}}" loading="lazy">⁠.</li>

<li>The <a href="Sine_and_cosine" title="Sine and cosine">sine and cosine</a> functions (⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin x}</annotation>

</semantics>

</math><img src="./09b4b55580d6a821a07ad9fe35be88976917b10b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.572ex; height:2.176ex;" alt="{\displaystyle \sin x}" loading="lazy">⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos x}</annotation>

</semantics>

</math><img src="./184ba70c3a71df25a25c09f34cd7f8175a9b5280.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.828ex; height:1.676ex;" alt="{\displaystyle \cos x}" loading="lazy">⁠) are continuous everywhere.</li>

<li>The <a href="Exponential_function" title="Exponential function">exponential function</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{x}}</annotation>

</semantics>

</math><img src="./841c0d168e64191c45a45e54c7e447defd17ec6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.256ex; height:2.343ex;" alt="{\displaystyle e^{x}}" loading="lazy">⁠ is continuous everywhere.</li>

<li>The <a href="Natural_logarithm" title="Natural logarithm">natural logarithm</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ln x}</annotation>

</semantics>

</math><img src="./ed172b0f5195382a3500c713941f945ad4db3898.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.656ex; height:2.176ex;" alt="{\displaystyle \ln x}" loading="lazy">⁠ is continuous on the domain formed by all positive real numbers ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\mid x>0\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{x\mid x>0\}}</annotation>

</semantics>

</math><img src="./60c3d1f6d96471ae2fd8d7cde451468ca42329c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.182ex; height:2.843ex;" alt="{\displaystyle \{x\mid x>0\}}" loading="lazy">⁠.</li></ul>

These rules imply that every <a href="Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial function</a> is continuous everywhere and that a <a href="Rational_function" title="Rational function">rational function</a> is continuous everywhere where it is defined, if the numerator and the denominator have no common <a href="Zero_of_a_function" title="Zero of a function">zeros</a>. More generally, the quotient of two continuous functions is continuous outside the zeros of the denominator.

An example of a function for which the above rules are not sufficient is the <a href="Sinc_function" title="Sinc function">sinc function</a>, which is defined by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sinc} (0)=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sinc} (0)=1}</annotation>

</semantics>

</math><img src="./adf163a388f638c8b04647ae06a0c4a07a5f2843.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.121ex; height:2.843ex;" alt="{\displaystyle \operatorname {sinc} (0)=1}" loading="lazy">⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sinc} (x)={\tfrac {\sin x}{x}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sinc} (x)={\tfrac {\sin x}{x}}}</annotation>

</semantics>

</math><img src="./989e87f89540c087649ac578213cab21aa183140.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.308ex; height:3.343ex;" alt="{\displaystyle \operatorname {sinc} (x)={\tfrac {\sin x}{x}}}" loading="lazy">⁠ for ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\neq 0}</annotation>

</semantics>

</math><img src="./35a455db7b2aab1b0e72ccbc7385e4424e2372e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle x\neq 0}" loading="lazy">⁠. The above rules show immediately that the function is continuous for ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\neq 0}</annotation>

</semantics>

</math><img src="./35a455db7b2aab1b0e72ccbc7385e4424e2372e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle x\neq 0}" loading="lazy">⁠, but, for proving the continuity at ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">⁠, one has to prove

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1.}</annotation>

</semantics>

</math>

As this is true, one gets that the sinc function is continuous function on all real numbers.

<div class="mw-heading mw-heading3"><h3 id="Examples_of_discontinuous_functions">Examples of discontinuous functions</h3></div>

An example of a discontinuous function is the <a href="Heaviside_step_function" title="Heaviside step function">Heaviside step function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy">, defined by

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(x)={\begin{cases}1&{\text{ if }}x\geq 0\\0&{\text{ if }}x<0\end{cases}}}</annotation>

</semantics>

</math>

Pick for instance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon =1/2}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon =1/2}</annotation>

</semantics>

</math><img src="./6018026f8800d4722329d039a6026e0c92e12297.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.669ex; height:2.843ex;" alt="{\displaystyle \varepsilon =1/2}" loading="lazy">. Then there is no <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta }</annotation>

</semantics>

</math><img src="./c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy">-neighborhood around <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=0}</annotation>

</semantics>

</math><img src="./953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}" loading="lazy">, i.e. no open interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\delta ,\;\delta )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (-\delta ,\;\delta )}</annotation>

</semantics>

</math><img src="./bfbd5b27b26d818200d19b5c1458ab932ea5a982.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.394ex; height:2.843ex;" alt="{\displaystyle (-\delta ,\;\delta )}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta >0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta >0,}</annotation>

</semantics>

</math><img src="./60f776a7a2322f7eb139801d00d029887455b081.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.956ex; height:2.676ex;" alt="{\displaystyle \delta >0,}" loading="lazy"> that will force all the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(x)}</annotation>

</semantics>

</math><img src="./11f241aa7195bebab9d0a3c248ea97ef0c78b1ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.203ex; height:2.843ex;" alt="{\displaystyle H(x)}" loading="lazy"> values to be within the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

</semantics>

</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy">-neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(0)}</annotation>

</semantics>

</math><img src="./f9262ffa8e12658e600949657d6d05927b86dd12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.035ex; height:2.843ex;" alt="{\displaystyle H(0)}" loading="lazy">, i.e. within <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1/2,\;3/2)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1/2,\;3/2)}</annotation>

</semantics>

</math><img src="./7a475e0bfe422618492ac2b69a003875a2a1661d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.463ex; height:2.843ex;" alt="{\displaystyle (1/2,\;3/2)}" loading="lazy">. Intuitively, we can think of this type of discontinuity as a sudden <a href="Jump_discontinuity" class="mw-redirect" title="Jump discontinuity">jump</a> in function values.

Similarly, the <a href="Sign_function" title="Sign function">signum</a> or sign function

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

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</mtd>

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<mtr>

<mtd>

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<mtd>

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</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(x)={\begin{cases}\;\;\ 1&{\text{ if }}x>0\\\;\;\ 0&{\text{ if }}x=0\\-1&{\text{ if }}x<0\end{cases}}}</annotation>

</semantics>

</math>

is discontinuous at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=0}</annotation>

</semantics>

</math><img src="./953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}" loading="lazy"> but continuous everywhere else. Yet another example: the function

<mrow>

<mtr>

<mtd>

<mrow>

<msup>

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</msup>

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<mtd>

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<mtr>

<mtd>

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<mtd>

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</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)={\begin{cases}\sin \left(x^{-2}\right)&{\text{ if }}x\neq 0\\0&{\text{ if }}x=0\end{cases}}}</annotation>

</semantics>

</math>

is continuous everywhere apart from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=0}</annotation>

</semantics>

</math><img src="./953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}" loading="lazy">.

Besides plausible continuities and discontinuities like above, there are also functions with a behavior, often coined <a href="Pathological_(mathematics)" title="Pathological (mathematics)">pathological</a>, for example, <a href="Thomae's_function" title="Thomae's function">Thomae's function</a>,

<mrow>

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<mtd>

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<mtd>

<mfrac>

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<mtd>

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<mfrac>

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<mtext>(in lowest terms) is a rational number</mtext>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

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<mtext> is irrational</mtext>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)={\begin{cases}1&{\text{ if }}x=0\\{\frac {1}{q}}&{\text{ if }}x={\frac {p}{q}}{\text{(in lowest terms) is a rational number}}\\0&{\text{ if }}x{\text{ is irrational}}.\end{cases}}}</annotation>

</semantics>

</math>

is continuous at all irrational numbers and discontinuous at all rational numbers. In a similar vein, <a href="Dirichlet's_function" class="mw-redirect" title="Dirichlet's function">Dirichlet's function</a>, the <a href="Indicator_function" title="Indicator function">indicator function</a> for the set of rational numbers,

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

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<mtext> is irrational </mtext>

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</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

<mtext> is rational </mtext>

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</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D(x)={\begin{cases}0&{\text{ if }}x{\text{ is irrational }}(\in \mathbb {R} \setminus \mathbb {Q} )\\1&{\text{ if }}x{\text{ is rational }}(\in \mathbb {Q} )\end{cases}}}</annotation>

</semantics>

</math>

is nowhere continuous.

<div class="mw-heading mw-heading3"><h3 id="Properties">Properties</h3></div>

<div class="mw-heading mw-heading4"><h4 id="A_useful_lemma">A useful lemma</h4></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> be a function that is continuous at a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0},}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0},}</annotation>

</semantics>

</math><img src="./c8b35dd572e629881da4083ad1681bc7cf420304.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.031ex; height:2.009ex;" alt="{\displaystyle x_{0},}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{0}}</annotation>

</semantics>

</math><img src="./6d943dbbb0b56ca750c4d62c5b54b4ae29a773da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{0}}" loading="lazy"> be a value such <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(x_{0}\right)\neq y_{0}.}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\left(x_{0}\right)\neq y_{0}.}</annotation>

</semantics>

</math><img src="./c08dd1ffd96045e136ddc85b00310cbfac96b4f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.798ex; height:2.843ex;" alt="{\displaystyle f\left(x_{0}\right)\neq y_{0}.}" loading="lazy"> Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\neq y_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)\neq y_{0}}</annotation>

</semantics>

</math><img src="./4e0fa541681eff5e18b5daffdc2b0789e5c936cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.709ex; height:2.843ex;" alt="{\displaystyle f(x)\neq y_{0}}" loading="lazy"> throughout some neighbourhood of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}.}</annotation>

</semantics>

</math><img src="./7ec756993d89cc1bd74f84040c07f5e11f0a8102.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.031ex; height:2.009ex;" alt="{\displaystyle x_{0}.}" loading="lazy"><a href="#cite_note-13">[13]</a>

Proof: By the definition of continuity, take <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon ={\frac {|y_{0}-f(x_{0})|}{2}}>0}">

<mfrac>

<mrow>

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<msub>

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<msub>

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</msub>

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</mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon ={\frac {|y_{0}-f(x_{0})|}{2}}>0}</annotation>

</semantics>

</math><img src="./a77ab76950a87ad1bdd26afabb5c0e5b48b683f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.078ex; height:5.676ex;" alt="{\displaystyle \varepsilon ={\frac {|y_{0}-f(x_{0})|}{2}}>0}" loading="lazy"> , then there exists <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta >0}</annotation>

</semantics>

</math><img src="./595d5cea06fdcaf2642caf549eda2cfc537958a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta >0}" loading="lazy"> such that

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mfrac>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mfrac>

</mrow>

<mtext> whenever </mtext>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left|f(x)-f(x_{0})\right|<{\frac {\left|y_{0}-f(x_{0})\right|}{2}}\quad {\text{ whenever }}\quad |x-x_{0}|<\delta }</annotation>

</semantics>

</math>

Suppose there is a point in the neighbourhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x-x_{0}|<\delta }">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |x-x_{0}|<\delta }</annotation>

</semantics>

</math><img src="./b2f902c12a7f7e5be35218a7d6f5cde43e036d86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.995ex; height:2.843ex;" alt="{\displaystyle |x-x_{0}|<\delta }" loading="lazy"> for which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=y_{0};}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=y_{0};}</annotation>

</semantics>

</math><img src="./6540e68d80c522f65f5d6eb1c2c60fc60501787c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.356ex; height:2.843ex;" alt="{\displaystyle f(x)=y_{0};}" loading="lazy"> then we have the contradiction

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mfrac>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left|f(x_{0})-y_{0}\right|<{\frac {\left|f(x_{0})-y_{0}\right|}{2}}.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading4"><h4 id="Intermediate_value_theorem">Intermediate value theorem</h4></div>

The <a href="Intermediate_value_theorem" title="Intermediate value theorem">intermediate value theorem</a> is an <a href="Existence_theorem" title="Existence theorem">existence theorem</a>, based on the real number property of <a href="Real_number#Completeness" title="Real number">completeness</a>, and states:

<dl><dd>If the real-valued function f is continuous on the <a href="Interval_(mathematics)" title="Interval (mathematics)">closed interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b],}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [a,b],}</annotation>

</semantics>

</math><img src="./2d493b840f8326ba81ff9d95b4edf1effd5f2842.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.202ex; height:2.843ex;" alt="{\displaystyle [a,b],}" loading="lazy"> and k is some number between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(a)}</annotation>

</semantics>

</math><img src="./368cb4b81ba5754d7a354a4ce49c2f1084bdaace.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.318ex; height:2.843ex;" alt="{\displaystyle f(a)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(b),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(b),}</annotation>

</semantics>

</math><img src="./2540a015cb0efa1fedb85a700dafc3f666d74192.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.732ex; height:2.843ex;" alt="{\displaystyle f(b),}" loading="lazy"> then there is some number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in [a,b],}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c\in [a,b],}</annotation>

</semantics>

</math><img src="./657c696455d34f8f86aad0515088771fe0d1f229.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.049ex; height:2.843ex;" alt="{\displaystyle c\in [a,b],}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(c)=k.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(c)=k.}</annotation>

</semantics>

</math><img src="./a1a98406c8be7cd52bafc8bcb82980dcb7723bbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.051ex; height:2.843ex;" alt="{\displaystyle f(c)=k.}" loading="lazy"></dd></dl>

For example, if a child grows from 1 m to 1.5 m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25 m.

As a consequence, if f is continuous on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>

</semantics>

</math><img src="./9c4b788fc5c637e26ee98b45f89a5c08c85f7935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(a)}</annotation>

</semantics>

</math><img src="./368cb4b81ba5754d7a354a4ce49c2f1084bdaace.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.318ex; height:2.843ex;" alt="{\displaystyle f(a)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(b)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(b)}</annotation>

</semantics>

</math><img src="./adbef9aaf500c30a3f107e5d7efb7a8627d3bba9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.085ex; height:2.843ex;" alt="{\displaystyle f(b)}" loading="lazy"> differ in <a href="Sign_(mathematics)" title="Sign (mathematics)">sign</a>, then, at some point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in [a,b],}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c\in [a,b],}</annotation>

</semantics>

</math><img src="./657c696455d34f8f86aad0515088771fe0d1f229.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.049ex; height:2.843ex;" alt="{\displaystyle c\in [a,b],}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(c)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(c)}</annotation>

</semantics>

</math><img src="./97d4cbf97e497bd1b20228d6a1c12f5f34b87213.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.095ex; height:2.843ex;" alt="{\displaystyle f(c)}" loading="lazy"> must equal <a href="0_(number)" class="mw-redirect" title="0 (number)">zero</a>.

<div class="mw-heading mw-heading4"><h4 id="Extreme_value_theorem">Extreme value theorem</h4></div>

The <a href="Extreme_value_theorem" title="Extreme value theorem">extreme value theorem</a> states that if a function f is defined on a closed interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>

</semantics>

</math><img src="./9c4b788fc5c637e26ee98b45f89a5c08c85f7935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}" loading="lazy"> (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in [a,b]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c\in [a,b]}</annotation>

</semantics>

</math><img src="./997256364b06acf0710e5d24da39e8c42991a249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.402ex; height:2.843ex;" alt="{\displaystyle c\in [a,b]}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(c)\geq f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(c)\geq f(x)}</annotation>

</semantics>

</math><img src="./2a66e4bbfb78b9a5021210f3e0a23177f930afe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.611ex; height:2.843ex;" alt="{\displaystyle f(c)\geq f(x)}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in [a,b].}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in [a,b].}</annotation>

</semantics>

</math><img src="./584ca2ac74e6072c1f7e88ab8f79d523ff8e22a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.372ex; height:2.843ex;" alt="{\displaystyle x\in [a,b].}" loading="lazy"> The same is true of the minimum of f. These statements are not, in general, true if the function is defined on an open interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a,b)}</annotation>

</semantics>

</math><img src="./d7e5710198f33b00695903460983021e75860e2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}" loading="lazy"> (or any set that is not both closed and bounded), as, for example, the continuous function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {1}{x}},}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{x}},}</annotation>

</semantics>

</math><img src="./c9fe877f6ddcb0b0991bdfd94f49f191d67660d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.329ex; height:5.176ex;" alt="{\displaystyle f(x)={\frac {1}{x}},}" loading="lazy"> defined on the open interval (0,1), does not attain a maximum, being unbounded above.

<div class="mw-heading mw-heading4"><h4 id="Relation_to_differentiability_and_integrability">Relation to differentiability and integrability</h4></div>

Every <a href="Differentiable_function" title="Differentiable function">differentiable function</a>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:(a,b)\to \mathbb {R} }</annotation>

</semantics>

</math>

is continuous, as can be shown. The <a href="Theorem#Converse" title="Theorem">converse</a> does not hold: for example, the <a href="Absolute_value" title="Absolute value">absolute value</a> function

</mrow>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

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<mtd>

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<annotation encoding="application/x-tex">{\displaystyle f(x)=|x|={\begin{cases}\;\;\ x&{\text{ if }}x\geq 0\\-x&{\text{ if }}x<0\end{cases}}}</annotation>

</semantics>

</math><img src="./c25bd36f728d79761109bac016db60c7a91370ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.366ex; height:6.176ex;" alt="{\displaystyle f(x)=|x|={\begin{cases}\;\;\ x&{\text{ if }}x\geq 0\\-x&{\text{ if }}x<0\end{cases}}}" loading="lazy"></dd></dl>

is everywhere continuous. However, it is not differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x=0}</annotation>

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</math><img src="./953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}" loading="lazy"> (but is so everywhere else). <a href="Weierstrass_function" title="Weierstrass function">Weierstrass's function</a> is also everywhere continuous but nowhere differentiable.

The <a href="Derivative" title="Derivative">derivative</a> f′(x) of a differentiable function f(x) need not be continuous. If f′(x) is continuous, f(x) is said to be continuously differentiable. The set of such functions is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{1}((a,b)).}">

<msup>

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</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{1}((a,b)).}</annotation>

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</math><img src="./82080e935a0c6b99c9d0b3d93b6f585c4c13f97b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.379ex; height:3.176ex;" alt="{\displaystyle C^{1}((a,b)).}" loading="lazy"> More generally, the set of functions

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:\Omega \to \mathbb {R} }</annotation>

</semantics>

</math>

(from an open interval (or <a href="Open_subset" class="mw-redirect" title="Open subset">open subset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"> to the reals) such that f is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> times differentiable and such that the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-th derivative of f is continuous is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{n}(\Omega ).}">

<msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{n}(\Omega ).}</annotation>

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</math><img src="./618a2b90010e9314cd36213cd47d5f6353c9aa72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.15ex; height:2.843ex;" alt="{\displaystyle C^{n}(\Omega ).}" loading="lazy"> See <a href="Differentiability_class" class="mw-redirect" title="Differentiability class">differentiability class</a>. In the field of computer graphics, properties related (but not identical) to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{0},C^{1},C^{2}}">

<msup>

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</msup>

<msup>

</mrow>

</msup>

<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{0},C^{1},C^{2}}</annotation>

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</math><img src="./b5b8d30e98472f6a74dfeb577609b6bfe36b51fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.624ex; height:3.009ex;" alt="{\displaystyle C^{0},C^{1},C^{2}}" loading="lazy"> are sometimes called <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{0}}">

<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle G^{0}}</annotation>

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</math><img src="./7c332d09499538333927fb909f9e38cfc991ada3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.881ex; height:2.676ex;" alt="{\displaystyle G^{0}}" loading="lazy"> (continuity of position), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G^{1}}</annotation>

</semantics>

</math><img src="./d9a781acdc595d4a41f40377b194350c27ef8529.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.881ex; height:2.676ex;" alt="{\displaystyle G^{1}}" loading="lazy"> (continuity of tangency), and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G^{2}}</annotation>

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</math><img src="./ecc876d9f46199f549440f26bda145d167c76b4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.881ex; height:2.676ex;" alt="{\displaystyle G^{2}}" loading="lazy"> (continuity of curvature); see <a href="Smoothness#Smoothness_of_curves_and_surfaces" title="Smoothness">Smoothness of curves and surfaces</a>.

Every continuous function

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle f:[a,b]\to \mathbb {R} }</annotation>

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is <a href="Integrable_function" class="mw-redirect" title="Integrable function">integrable</a> (for example in the sense of the <a href="Riemann_integral" title="Riemann integral">Riemann integral</a>). The converse does not hold, as the (integrable but discontinuous) <a href="Sign_function" title="Sign function">sign function</a> shows.

<div class="mw-heading mw-heading4"><h4 id="Pointwise_and_uniform_limits">Pointwise and uniform limits</h4></div>

Given a <a href="Sequence_(mathematics)" class="mw-redirect" title="Sequence (mathematics)">sequence</a>

<msub>

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</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle f_{1},f_{2},\dotsc :I\to \mathbb {R} }</annotation>

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</math>

of functions such that the limit

</mrow>

</munder>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle f(x):=\lim _{n\to \infty }f_{n}(x)}</annotation>

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</math>

exists for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in D,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in D,}</annotation>

</semantics>

</math><img src="./5c4bc1c9d9514af5201950067c06acc1ca8f6151.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.741ex; height:2.509ex;" alt="{\displaystyle x\in D,}" loading="lazy">, the resulting function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

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</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> is referred to as the <a href="Pointwise_convergence" title="Pointwise convergence">pointwise limit</a> of the sequence of functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(f_{n}\right)_{n\in N}.}">

<msub>

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<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(f_{n}\right)_{n\in N}.}</annotation>

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</math><img src="./b7e6012ee43229264781af316f20b31c626cb656.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.588ex; height:3.009ex;" alt="{\displaystyle \left(f_{n}\right)_{n\in N}.}" loading="lazy"> The pointwise limit function need not be continuous, even if all functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{n}}</annotation>

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</math><img src="./b2702450f0458a5e01a698e248af552a7fab2b50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:2.509ex;" alt="{\displaystyle f_{n}}" loading="lazy"> are continuous, as the animation at the right shows. However, f is continuous if all functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{n}}</annotation>

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</math><img src="./b2702450f0458a5e01a698e248af552a7fab2b50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:2.509ex;" alt="{\displaystyle f_{n}}" loading="lazy"> are continuous and the sequence <a href="Uniform_convergence" title="Uniform convergence">converges uniformly</a>, by the <a href="Uniform_convergence_theorem" class="mw-redirect" title="Uniform convergence theorem">uniform convergence theorem</a>. This theorem can be used to show that the <a href="Exponential_function" title="Exponential function">exponential functions</a>, <a href="Logarithm" title="Logarithm">logarithms</a>, <a href="Square_root" title="Square root">square root</a> function, and <a href="Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a> are continuous.

<div class="mw-heading mw-heading3"><h3 id="Directional_Continuity">Directional Continuity</h3></div>

<div class="gallerytext">A right-continuous function</div>

</li>

<div class="gallerytext">A left-continuous function</div>

</li>

</ul></div>

Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right and left continuous functions) and <a href="Semi-continuity" title="Semi-continuity">semi-continuity</a>. Roughly speaking, a function is right-continuous if no jump occurs when the limit point is approached from the right. Formally, f is said to be right-continuous at the point c if the following holds: For any number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation>

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</math><img src="./e04ec3670b50384a3ce48aca42e7cc5131a06b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}" loading="lazy"> however small, there exists some number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta >0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \delta >0}</annotation>

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</math><img src="./595d5cea06fdcaf2642caf549eda2cfc537958a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta >0}" loading="lazy"> such that for all x in the domain with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c<x<c+\delta ,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c<x<c+\delta ,}</annotation>

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</math><img src="./75ebdcebca47b67680580d3b67c593380ccc8969.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.076ex; height:2.676ex;" alt="{\displaystyle c<x<c+\delta ,}" loading="lazy"> the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

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</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> satisfies

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<annotation encoding="application/x-tex">{\displaystyle |f(x)-f(c)|<\varepsilon }</annotation>

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</math>

This is the same condition as continuous functions, except it is required to hold only for x strictly larger than c. Requiring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(x)-f(c)|<\varepsilon }">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle |f(x)-f(c)|<\varepsilon }</annotation>

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</math><img src="./d6294a72a7917cdfb9afcd2d242e879e3a3f6261.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.828ex; height:2.843ex;" alt="{\displaystyle |f(x)-f(c)|<\varepsilon }" loading="lazy"> to hold instead for all x with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c-\delta <x<c}">

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<annotation encoding="application/x-tex">{\displaystyle c-\delta <x<c}</annotation>

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</math><img src="./cd9ab2b86671c949fe1e0e3a7f220c1d0e8d1f7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.429ex; height:2.509ex;" alt="{\displaystyle c-\delta <x<c}" loading="lazy"> yields the notion of left-continuous functions. A function is continuous if and only if it is both right-continuous and left-continuous.

<div class="mw-heading mw-heading3"><h3 id="Semicontinuity">Semicontinuity</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Semicontinuity" class="mw-redirect" title="Semicontinuity">Semicontinuity</a></div>

A function f is <a href="Semi-continuity" title="Semi-continuity">lower semi-continuous</a> at the point c if, roughly, any jumps that might occur only go down, but not up. That is, for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0,}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon >0,}</annotation>

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</math><img src="./4d780d8dff4b26013c7d5d0efbc1acb92b60645b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.991ex; height:2.509ex;" alt="{\displaystyle \varepsilon >0,}" loading="lazy"> there exists some number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta >0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \delta >0}</annotation>

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</math><img src="./595d5cea06fdcaf2642caf549eda2cfc537958a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta >0}" loading="lazy"> such that for all x in the domain with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x-c|<\delta ,}">

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<annotation encoding="application/x-tex">{\displaystyle |x-c|<\delta ,}</annotation>

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</math><img src="./a4c1f32c4aea3a40f333561b0f1f89da640d1b5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.265ex; height:2.843ex;" alt="{\displaystyle |x-c|<\delta ,}" loading="lazy"> the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

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</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> satisfies

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<annotation encoding="application/x-tex">{\displaystyle f(x)\geq f(c)-\epsilon .}</annotation>

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</math>

The reverse condition is <a href="Semi-continuity" title="Semi-continuity">upper semi-continuity</a>.

<div class="mw-heading mw-heading2"><h2 id="Continuous_functions_between_metric_spaces">Continuous functions between metric spaces</h2></div>

The concept of continuous real-valued functions can be generalized to functions between <a href="Metric_space" title="Metric space">metric spaces</a>. A metric space is a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> equipped with a function (called <a href="Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{X},}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle d_{X},}</annotation>

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</math><img src="./77cbd2d20111212f9d91b0189ca2516e7a50d695.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.488ex; height:2.509ex;" alt="{\displaystyle d_{X},}" loading="lazy"> that can be thought of as a measurement of the distance of any two elements in X. Formally, the metric is a function

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<annotation encoding="application/x-tex">{\displaystyle d_{X}:X\times X\to \mathbb {R} }</annotation>

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that satisfies a number of requirements, notably the <a href="Triangle_inequality" title="Triangle inequality">triangle inequality</a>. Given two metric spaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(X,d_{X}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle \left(X,d_{X}\right)}</annotation>

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</math><img src="./6437ccc122e50a171123b3aa7761d4989c8d84bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.664ex; height:2.843ex;" alt="{\displaystyle \left(X,d_{X}\right)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(Y,d_{Y}\right)}">

<mrow>

<mrow>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \left(Y,d_{Y}\right)}</annotation>

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</math><img src="./9dac2c41e5665b3de3054b1e22ca36f8dd17273b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.312ex; height:2.843ex;" alt="{\displaystyle \left(Y,d_{Y}\right)}" loading="lazy"> and a function

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<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

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</math>

then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is continuous at the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c\in X}</annotation>

</semantics>

</math><img src="./20a6fd8987f71d0e8b6f844f05339748989a1267.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.828ex; height:2.176ex;" alt="{\displaystyle c\in X}" loading="lazy"> (with respect to the given metrics) if for any positive real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon >0,}</annotation>

</semantics>

</math><img src="./4d780d8dff4b26013c7d5d0efbc1acb92b60645b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.991ex; height:2.509ex;" alt="{\displaystyle \varepsilon >0,}" loading="lazy"> there exists a positive real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta >0}</annotation>

</semantics>

</math><img src="./595d5cea06fdcaf2642caf549eda2cfc537958a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta >0}" loading="lazy"> such that all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{X}(x,c)<\delta }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{X}(x,c)<\delta }</annotation>

</semantics>

</math><img src="./f9f46d52e6e7c28e83a3df7bff193764c977f407.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.168ex; height:2.843ex;" alt="{\displaystyle d_{X}(x,c)<\delta }" loading="lazy"> will also satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{Y}(f(x),f(c))<\varepsilon .}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{Y}(f(x),f(c))<\varepsilon .}</annotation>

</semantics>

</math><img src="./81fcc7958ee9f9182adc8a60c4b6b314b70ebb00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.879ex; height:2.843ex;" alt="{\displaystyle d_{Y}(f(x),f(c))<\varepsilon .}" loading="lazy"> As in the case of real functions above, this is equivalent to the condition that for every sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x_{n}\right)}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(x_{n}\right)}</annotation>

</semantics>

</math><img src="./3797df42d52cd3e009ad09ad9add4f5590fb3cba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.357ex; height:2.843ex;" alt="{\displaystyle \left(x_{n}\right)}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> with limit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim x_{n}=c,}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim x_{n}=c,}</annotation>

</semantics>

</math><img src="./b29c0a72e290d87e4bcf86fbaed283fb14ab4faf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.917ex; height:2.509ex;" alt="{\displaystyle \lim x_{n}=c,}" loading="lazy"> we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim f\left(x_{n}\right)=f(c).}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim f\left(x_{n}\right)=f(c).}</annotation>

</semantics>

</math><img src="./728158b6dcbdd61a197c99229cbd829a15bd7091.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.48ex; height:2.843ex;" alt="{\displaystyle \lim f\left(x_{n}\right)=f(c).}" loading="lazy"> The latter condition can be weakened as follows: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is continuous at the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> if and only if for every convergent sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x_{n}\right)}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(x_{n}\right)}</annotation>

</semantics>

</math><img src="./3797df42d52cd3e009ad09ad9add4f5590fb3cba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.357ex; height:2.843ex;" alt="{\displaystyle \left(x_{n}\right)}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> with limit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy">, the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(f\left(x_{n}\right)\right)}">

<mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(f\left(x_{n}\right)\right)}</annotation>

</semantics>

</math><img src="./87e1b0008e7b6b4a885c758657996d4d2a8ffeb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.832ex; height:2.843ex;" alt="{\displaystyle \left(f\left(x_{n}\right)\right)}" loading="lazy"> is a <a href="Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> is in the domain of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">.

The set of points at which a function between metric spaces is continuous is a <a href="G%CE%B4_set" title="Gδ set"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\delta }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{\delta }}</annotation>

</semantics>

</math><img src="./7b42b79015590af928aa28bda9514373aeac0e1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.8ex; height:2.509ex;" alt="{\displaystyle G_{\delta }}" loading="lazy"> set</a> – this follows from the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon -\delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon -\delta }</annotation>

</semantics>

</math><img src="./bdcdab850a3df0b5c13909e4b058fb35220f019d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.973ex; height:2.509ex;" alt="{\displaystyle \varepsilon -\delta }" loading="lazy"> definition of continuity.

This notion of continuity is applied, for example, in <a href="Functional_analysis" title="Functional analysis">functional analysis</a>. A key statement in this area says that a <a href="Linear_operator" class="mw-redirect" title="Linear operator">linear operator</a>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T:V\to W}</annotation>

</semantics>

</math>

between <a href="Normed_vector_space" title="Normed vector space">normed vector spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> (which are <a href="Vector_space" title="Vector space">vector spaces</a> equipped with a compatible <a href="Norm_(mathematics)" title="Norm (mathematics)">norm</a>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|x\|}</annotation>

</semantics>

</math><img src="./be45ae82237598ea927f321e2e1e04e36633b93b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle \|x\|}" loading="lazy">) is continuous if and only if it is <a href="Bounded_linear_operator" class="mw-redirect" title="Bounded linear operator">bounded</a>, that is, there is a constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> such that

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|T(x)\|\leq K\|x\|}</annotation>

</semantics>

</math>

for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in V.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in V.}</annotation>

</semantics>

</math><img src="./2db8a2c9b98774154e833074ffbab1d3fc6c0f61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.604ex; height:2.176ex;" alt="{\displaystyle x\in V.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Uniform,_Hölder_and_Lipschitz_continuity">Uniform, Hölder and Lipschitz continuity</h3></div>

The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta }</annotation>

</semantics>

</math><img src="./c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"> depends on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

</semantics>

</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"> and c in the definition above. Intuitively, a function f as above is <a href="Uniformly_continuous" class="mw-redirect" title="Uniformly continuous">uniformly continuous</a> if the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta }</annotation>

</semantics>

</math><img src="./c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"> does

not depend on the point c. More precisely, it is required that for every <a href="Real_number" title="Real number">real number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation>

</semantics>

</math><img src="./e04ec3670b50384a3ce48aca42e7cc5131a06b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}" loading="lazy"> there exists <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta >0}</annotation>

</semantics>

</math><img src="./595d5cea06fdcaf2642caf549eda2cfc537958a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta >0}" loading="lazy"> such that for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c,b\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c,b\in X}</annotation>

</semantics>

</math><img src="./fd571fa87c84b95b9a108164c662263bc8057d1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.859ex; height:2.509ex;" alt="{\displaystyle c,b\in X}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{X}(b,c)<\delta ,}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{X}(b,c)<\delta ,}</annotation>

</semantics>

</math><img src="./05cfa5267549dc0ee8838c1a7e9634f86099e762.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.483ex; height:2.843ex;" alt="{\displaystyle d_{X}(b,c)<\delta ,}" loading="lazy"> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{Y}(f(b),f(c))<\varepsilon .}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{Y}(f(b),f(c))<\varepsilon .}</annotation>

</semantics>

</math><img src="./7f1b4b40ec11d474d5bb4d0f7a8f0cc7d7763566.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.547ex; height:2.843ex;" alt="{\displaystyle d_{Y}(f(b),f(c))<\varepsilon .}" loading="lazy"> Thus, any uniformly continuous function is continuous. The converse does not generally hold but holds when the domain space X is <a href="Compact_topological_space" class="mw-redirect" title="Compact topological space">compact</a>. Uniformly continuous maps can be defined in the more general situation of <a href="Uniform_space" title="Uniform space">uniform spaces</a>.<a href="#cite_note-14">[14]</a>

A function is <a href="H%C3%B6lder_continuity" class="mw-redirect" title="Hölder continuity">Hölder continuous</a> with exponent α (a real number) if there is a constant K such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b,c\in X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b,c\in X,}</annotation>

</semantics>

</math><img src="./f54d4638b8e7c38193297cf5c3f5a9fd35e0718b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.506ex; height:2.509ex;" alt="{\displaystyle b,c\in X,}" loading="lazy"> the inequality

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{Y}(f(b),f(c))\leq K\cdot (d_{X}(b,c))^{\alpha }}</annotation>

</semantics>

</math>

holds. Any Hölder continuous function is uniformly continuous. The particular case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha =1}</annotation>

</semantics>

</math><img src="./03d67a45a44be8b8f15e99b7def2b0cf0aba1717.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha =1}" loading="lazy"> is referred to as <a href="Lipschitz_continuity" title="Lipschitz continuity">Lipschitz continuity</a>. That is, a function is Lipschitz continuous if there is a constant K such that the inequality

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{Y}(f(b),f(c))\leq K\cdot d_{X}(b,c)}</annotation>

</semantics>

</math>

holds for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b,c\in X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b,c\in X.}</annotation>

</semantics>

</math><img src="./8e1666d09b6ba1f1318839ce870fc11b114d551b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.506ex; height:2.509ex;" alt="{\displaystyle b,c\in X.}" loading="lazy"><a href="#cite_note-15">[15]</a> The Lipschitz condition occurs, for example, in the <a href="Picard%E2%80%93Lindel%C3%B6f_theorem" title="Picard–Lindelöf theorem">Picard–Lindelöf theorem</a> concerning the solutions of <a href="Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a>.

<div class="mw-heading mw-heading2"><h2 id="Continuous_functions_between_topological_spaces">Continuous functions between topological spaces</h2></div>

Another, more abstract, notion of continuity is the continuity of functions between <a href="Topological_space" title="Topological space">topological spaces</a> in which there generally is no formal notion of distance, as there is in the case of <a href="Metric_space" title="Metric space">metric spaces</a>. A topological space is a set X together with a topology on X, which is a set of <a href="Subset" title="Subset">subsets</a> of X satisfying a few requirements with respect to their unions and intersections that generalize the properties of the <a href="Open_ball" class="mw-redirect" title="Open ball">open balls</a> in metric spaces while still allowing one to talk about the <a href="Neighborhood_(mathematics)" class="mw-redirect" title="Neighborhood (mathematics)"> neighborhoods</a> of a given point. The elements of a topology are called <a href="Open_subset" class="mw-redirect" title="Open subset">open subsets</a> of X (with respect to the topology).

A function

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

</semantics>

</math>

between two topological spaces X and Y is continuous if for every open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\subseteq Y,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V\subseteq Y,}</annotation>

</semantics>

</math><img src="./6c783e5913932b26ab2d1696ede9ad24d763079e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.306ex; height:2.509ex;" alt="{\displaystyle V\subseteq Y,}" loading="lazy"> the <a href="Image_(mathematics)#Inverse_image" title="Image (mathematics)">inverse image</a>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(V)=\{x\in X\;|\;f(x)\in V\}}</annotation>

</semantics>

</math>

is an open subset of X. That is, f is a function between the sets X and Y (not on the elements of the topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{X}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{X}}</annotation>

</semantics>

</math><img src="./a94e80072681ec0d850d5d10601d58032619803e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.99ex; height:2.509ex;" alt="{\displaystyle T_{X}}" loading="lazy">), but the continuity of f depends on the topologies used on X and Y.

This is equivalent to the condition that the <a href="Image_(mathematics)#Inverse_image" title="Image (mathematics)">preimages</a> of the <a href="Closed_set" title="Closed set">closed sets</a> (which are the complements of the open subsets) in Y are closed in X.

An extreme example: if a set X is given the <a href="Discrete_topology" class="mw-redirect" title="Discrete topology">discrete topology</a> (in which every subset is open), all functions

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to T}</annotation>

</semantics>

</math>

to any topological space T are continuous. On the other hand, if X is equipped with the <a href="Indiscrete_topology" class="mw-redirect" title="Indiscrete topology">indiscrete topology</a> (in which the only open subsets are the empty set and X) and the space T set is at least <a href="T0_space" class="mw-redirect" title="T0 space">T0</a>, then the only continuous functions are the constant functions. Conversely, any function whose codomain is indiscrete is continuous.

<div class="mw-heading mw-heading3"><h3 id="Continuity_at_a_point">Continuity at a point</h3></div>

The translation in the language of neighborhoods of the <a href="(%CE%B5%2C_%CE%B4)-definition_of_limit" class="mw-redirect" title="(ε, δ)-definition of limit"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\varepsilon ,\delta )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\varepsilon ,\delta )}</annotation>

</semantics>

</math><img src="./08b05469628ac7ee747965d1cf3825d1f9861148.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.975ex; height:2.843ex;" alt="{\displaystyle (\varepsilon ,\delta )}" loading="lazy">-definition of continuity</a> leads to the following definition of the continuity at a point:

<blockquote class="quote-frame pullquote" style="font-size: 95%; padding: 0.5em 2em; background-color: var( --background-color-neutral-subtle, #f8f9fa ); color: var( --color-base, black ); border: 1px solid #aaa; display:table; float:none;"><div style="padding: 0.6em 1em;">A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> is continuous at a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> if and only if for any neighborhood V of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> in Y, there is a neighborhood U of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(U)\subseteq V.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(U)\subseteq V.}</annotation>

</semantics>

</math><img src="./fb7e2ee44ca195b6672f893f47f4e1b5b0cae043.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.403ex; height:2.843ex;" alt="{\displaystyle f(U)\subseteq V.}" loading="lazy"></div></blockquote>

This definition is equivalent to the same statement with neighborhoods restricted to open neighborhoods and can be restated in several ways by using <a href="Preimage" class="mw-redirect" title="Preimage">preimages</a> rather than images.

Also, as every set that contains a neighborhood is also a neighborhood, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(V)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(V)}</annotation>

</semantics>

</math><img src="./8b5171cc2674480baf68e55ff20378a5413d4728.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.25ex; height:3.176ex;" alt="{\displaystyle f^{-1}(V)}" loading="lazy"> is the largest subset U of X such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(U)\subseteq V,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(U)\subseteq V,}</annotation>

</semantics>

</math><img src="./730eee3ecf8f4c39fc15447566fccedb23282706.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.403ex; height:2.843ex;" alt="{\displaystyle f(U)\subseteq V,}" loading="lazy"> this definition may be simplified into:

<blockquote class="quote-frame pullquote" style="font-size: 95%; padding: 0.5em 2em; background-color: var( --background-color-neutral-subtle, #f8f9fa ); color: var( --color-base, black ); border: 1px solid #aaa; display:table; float:none;"><div style="padding: 0.6em 1em;">A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> is continuous at a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(V)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(V)}</annotation>

</semantics>

</math><img src="./8b5171cc2674480baf68e55ff20378a5413d4728.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.25ex; height:3.176ex;" alt="{\displaystyle f^{-1}(V)}" loading="lazy"> is a neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> for every neighborhood V of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> in Y.</div></blockquote>

As an open set is a set that is a neighborhood of all its points, a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> is continuous at every point of X if and only if it is a continuous function.

If X and Y are metric spaces, it is equivalent to consider the <a href="Neighborhood_system" class="mw-redirect" title="Neighborhood system">neighborhood system</a> of <a href="Open_ball" class="mw-redirect" title="Open ball">open balls</a> centered at x and f(x) instead of all neighborhoods. This gives back the above <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon -\delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon -\delta }</annotation>

</semantics>

</math><img src="./bdcdab850a3df0b5c13909e4b058fb35220f019d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.973ex; height:2.509ex;" alt="{\displaystyle \varepsilon -\delta }" loading="lazy"> definition of continuity in the context of metric spaces. In general topological spaces, there is no notion of nearness or distance. If, however, the target space is a <a href="Hausdorff_space" title="Hausdorff space">Hausdorff space</a>, it is still true that f is continuous at a if and only if the limit of f as x approaches a is f(a). At an isolated point, every function is continuous.

Given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation>

</semantics>

</math><img src="./e3ebdb0a09f0721ccdd0b779e0a21caf386be82a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" loading="lazy"> a map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> is continuous at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> if and only if whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation>

</semantics>

</math><img src="./e5622de88a69f68340f8dcb43d0b8bd443ba9e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}" loading="lazy"> is a filter on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> that <a href="Convergent_filter" class="mw-redirect" title="Convergent filter">converges</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> which is expressed by writing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}\to x,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}\to x,}</annotation>

</semantics>

</math><img src="./284e13e78dc1f10019185400bbcbde579ed8bab9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.134ex; height:2.509ex;" alt="{\displaystyle {\mathcal {B}}\to x,}" loading="lazy"> then necessarily <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f({\mathcal {B}})\to f(x)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f({\mathcal {B}})\to f(x)}</annotation>

</semantics>

</math><img src="./457b5f40fef05a407cdce5233450a5b355a8e07d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.663ex; height:2.843ex;" alt="{\displaystyle f({\mathcal {B}})\to f(x)}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y.}</annotation>

</semantics>

</math><img src="./0c668649af47a30006f93c9847d61fee8d9ffb61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" loading="lazy">

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}(x)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}(x)}</annotation>

</semantics>

</math><img src="./fa099930dd7af7c3339be12e975001ffb45e6241.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:5.476ex; height:3.009ex;" alt="{\displaystyle {\mathcal {N}}(x)}" loading="lazy"> denotes the <a href="Neighborhood_filter" class="mw-redirect" title="Neighborhood filter">neighborhood filter</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> is continuous at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f({\mathcal {N}}(x))\to f(x)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f({\mathcal {N}}(x))\to f(x)}</annotation>

</semantics>

</math><img src="./9c1138a603429a69d9e0ac6722048b984629a983.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.533ex; height:3.009ex;" alt="{\displaystyle f({\mathcal {N}}(x))\to f(x)}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y.}</annotation>

</semantics>

</math><img src="./0c668649af47a30006f93c9847d61fee8d9ffb61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" loading="lazy"><a href="#cite_note-FOOTNOTEDugundji1966211–221-16">[16]</a> Moreover, this happens if and only if the <a href="Prefilter" class="mw-redirect" title="Prefilter">prefilter</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f({\mathcal {N}}(x))}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f({\mathcal {N}}(x))}</annotation>

</semantics>

</math><img src="./10ca125cf36d48676c3e4df0c7bbec9a2c00fa9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.502ex; height:3.009ex;" alt="{\displaystyle f({\mathcal {N}}(x))}" loading="lazy"> is a <a href="Filter_base" class="mw-redirect" title="Filter base">filter base</a> for the neighborhood filter of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y.}</annotation>

</semantics>

</math><img src="./0c668649af47a30006f93c9847d61fee8d9ffb61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" loading="lazy"><a href="#cite_note-FOOTNOTEDugundji1966211–221-16">[16]</a>

<div class="mw-heading mw-heading3"><h3 id="Alternative_definitions">Alternative definitions</h3></div>

Several <a href="Characterizations_of_the_category_of_topological_spaces" class="mw-redirect" title="Characterizations of the category of topological spaces">equivalent definitions for a topological structure</a> exist; thus, several equivalent ways exist to define a continuous function.

<div class="mw-heading mw-heading4"><h4 id="Sequences_and_nets">Sequences and nets </h4></div>

In several contexts, the topology of a space is conveniently specified in terms of <a href="Limit_points" class="mw-redirect" title="Limit points">limit points</a>. This is often accomplished by specifying when a point is the <a href="Limit_of_a_sequence" title="Limit of a sequence">limit of a sequence</a>. Still, for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points <a href="Indexed_family" title="Indexed family">indexed</a> by a <a href="Directed_set" title="Directed set">directed set</a>, known as <a href="Net_(mathematics)" title="Net (mathematics)">nets</a>. A function is (Heine-)continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition.

In detail, a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> is <a href="Sequential_continuity" class="mw-redirect" title="Sequential continuity">sequentially continuous</a> if whenever a sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x_{n}\right)}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(x_{n}\right)}</annotation>

</semantics>

</math><img src="./3797df42d52cd3e009ad09ad9add4f5590fb3cba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.357ex; height:2.843ex;" alt="{\displaystyle \left(x_{n}\right)}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> converges to a limit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x,}</annotation>

</semantics>

</math><img src="./feff4d40084c7351bf57b11ba2427f6331f5bdbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}" loading="lazy"> the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(f\left(x_{n}\right)\right)}">

<mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(f\left(x_{n}\right)\right)}</annotation>

</semantics>

</math><img src="./87e1b0008e7b6b4a885c758657996d4d2a8ffeb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.832ex; height:2.843ex;" alt="{\displaystyle \left(f\left(x_{n}\right)\right)}" loading="lazy"> converges to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x).}</annotation>

</semantics>

</math><img src="./e889a7f3ee0216318cbf9f3478208fa1404cc51c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.064ex; height:2.843ex;" alt="{\displaystyle f(x).}" loading="lazy"> Thus, sequentially continuous functions "preserve sequential limits." Every continuous function is sequentially continuous. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a <a href="First-countable_space" title="First-countable space">first-countable space</a> and <a href="Axiom_of_countable_choice" title="Axiom of countable choice">countable choice</a> holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a metric space, sequential continuity and continuity are equivalent. For non-first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called <a href="Sequential_space" title="Sequential space">sequential spaces</a>.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve the limits of nets, and this property characterizes continuous functions.

For instance, consider the case of real-valued functions of one real variable:<a href="#cite_note-17">[17]</a>

/* start https://en.wikipedia.org/ */

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</style><div class="math_theorem" style="">

Theorem—A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:A\subseteq \mathbb {R} \to \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:A\subseteq \mathbb {R} \to \mathbb {R} }</annotation>

</semantics>

</math><img src="./bfac900d2166be1c5e2ff22179bbf3fcfd360c6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.028ex; height:2.509ex;" alt="{\displaystyle f:A\subseteq \mathbb {R} \to \mathbb {R} }" loading="lazy"> is continuous at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> if and only if it is <a href="Sequentially_continuous" class="mw-redirect" title="Sequentially continuous">sequentially continuous</a> at that point.

</div>

/* start https://en.wikipedia.org/ */

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/* end https://en.wikipedia.org/ */

</style>

<th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:left;"><div style="font-size:115%;">Proof</div>

</th></tr>

<tr>

Proof. Assume that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:A\subseteq \mathbb {R} \to \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:A\subseteq \mathbb {R} \to \mathbb {R} }</annotation>

</semantics>

</math><img src="./bfac900d2166be1c5e2ff22179bbf3fcfd360c6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.028ex; height:2.509ex;" alt="{\displaystyle f:A\subseteq \mathbb {R} \to \mathbb {R} }" loading="lazy"> is continuous at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> (in the sense of <a href="(%CE%B5%2C_%CE%B4)-definition_of_limit" class="mw-redirect" title="(ε, δ)-definition of limit"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon -\delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \epsilon -\delta }</annotation>

</semantics>

</math><img src="./6b6a3dac265e0e391447286d7b6f274496363747.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.833ex; height:2.509ex;" alt="{\displaystyle \epsilon -\delta }" loading="lazy"> continuity</a>). Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x_{n}\right)_{n\geq 1}}">

<msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(x_{n}\right)_{n\geq 1}}</annotation>

</semantics>

</math><img src="./974dbe2566f9cc8d47617c84a95e921d47c31a4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.676ex; height:3.176ex;" alt="{\displaystyle \left(x_{n}\right)_{n\geq 1}}" loading="lazy"> be a sequence converging at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> (such a sequence always exists, for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}=x,{\text{ for all }}n}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n}=x,{\text{ for all }}n}</annotation>

</semantics>

</math><img src="./3749e6768c92848ba5a90eff4c4fd5e48c022e53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.389ex; height:2.509ex;" alt="{\displaystyle x_{n}=x,{\text{ for all }}n}" loading="lazy">); since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is continuous at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy">

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \epsilon >0\,\exists \delta _{\epsilon }>0:0<|x-x_{0}|<\delta _{\epsilon }\implies |f(x)-f(x_{0})|<\epsilon .\quad (*)}">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \forall \epsilon >0\,\exists \delta _{\epsilon }>0:0<|x-x_{0}|<\delta _{\epsilon }\implies |f(x)-f(x_{0})|<\epsilon .\quad (*)}</annotation>

</semantics>

</math>

For any such <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\epsilon }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{\epsilon }}</annotation>

</semantics>

</math><img src="./24e968af8ad9cb12efa4cf3a983c7766b0606f49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.932ex; height:2.676ex;" alt="{\displaystyle \delta _{\epsilon }}" loading="lazy"> we can find a natural number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{\epsilon }>0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu _{\epsilon }>0}</annotation>

</semantics>

</math><img src="./9aa81b04bcd646592a4c205d60c223b224abb0bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.309ex; height:2.509ex;" alt="{\displaystyle \nu _{\epsilon }>0}" loading="lazy"> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>\nu _{\epsilon },}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n>\nu _{\epsilon },}</annotation>

</semantics>

</math><img src="./1995ad229db9eb99a33cc074ec4dede59ea1874b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.188ex; height:2.176ex;" alt="{\displaystyle n>\nu _{\epsilon },}" loading="lazy">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |x_{n}-x_{0}|<\delta _{\epsilon },}</annotation>

</semantics>

</math>

since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x_{n}\right)}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(x_{n}\right)}</annotation>

</semantics>

</math><img src="./3797df42d52cd3e009ad09ad9add4f5590fb3cba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.357ex; height:2.843ex;" alt="{\displaystyle \left(x_{n}\right)}" loading="lazy"> converges at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy">; combining this with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (*)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (*)}</annotation>

</semantics>

</math><img src="./fae74f24018ce1ff0dc698dbe8181555eb4cd768.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle (*)}" loading="lazy"> we obtain

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \epsilon >0\,\exists \nu _{\epsilon }>0:\forall n>\nu _{\epsilon }\quad |f(x_{n})-f(x_{0})|<\epsilon .}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \forall \epsilon >0\,\exists \nu _{\epsilon }>0:\forall n>\nu _{\epsilon }\quad |f(x_{n})-f(x_{0})|<\epsilon .}</annotation>

</semantics>

</math>

Assume on the contrary that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is sequentially continuous and proceed by contradiction: suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is not continuous at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy">

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists \epsilon >0:\forall \delta _{\epsilon }>0,\,\exists x_{\delta _{\epsilon }}:0<|x_{\delta _{\epsilon }}-x_{0}|<\delta _{\epsilon }\implies |f(x_{\delta _{\epsilon }})-f(x_{0})|>\epsilon }">

<msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mrow>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exists \epsilon >0:\forall \delta _{\epsilon }>0,\,\exists x_{\delta _{\epsilon }}:0<|x_{\delta _{\epsilon }}-x_{0}|<\delta _{\epsilon }\implies |f(x_{\delta _{\epsilon }})-f(x_{0})|>\epsilon }</annotation>

</semantics>

</math>

then we can take <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\epsilon }=1/n,\,\forall n>0}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{\epsilon }=1/n,\,\forall n>0}</annotation>

</semantics>

</math><img src="./ea8e54f5a88a652faac11ff7e9e75f3ec2c0edb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.12ex; height:2.843ex;" alt="{\displaystyle \delta _{\epsilon }=1/n,\,\forall n>0}" loading="lazy"> and call the corresponding point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\delta _{\epsilon }}=:x_{n}}">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{\delta _{\epsilon }}=:x_{n}}</annotation>

</semantics>

</math><img src="./fa02a24b471d66ac51665b910c5ff31bcadac309.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.292ex; height:2.176ex;" alt="{\displaystyle x_{\delta _{\epsilon }}=:x_{n}}" loading="lazy">: in this way we have defined a sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{n})_{n\geq 1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{n})_{n\geq 1}}</annotation>

</semantics>

</math><img src="./9da3a94a228d4e7a6fe0fc81adfc8c92795068d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.676ex; height:2.843ex;" alt="{\displaystyle (x_{n})_{n\geq 1}}" loading="lazy"> such that

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall n>0\quad |x_{n}-x_{0}|<{\frac {1}{n}},\quad |f(x_{n})-f(x_{0})|>\epsilon }">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \forall n>0\quad |x_{n}-x_{0}|<{\frac {1}{n}},\quad |f(x_{n})-f(x_{0})|>\epsilon }</annotation>

</semantics>

</math>

by construction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}\to x_{0}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n}\to x_{0}}</annotation>

</semantics>

</math><img src="./91d659f7e8420bcc79c0f1c0a1979f969c52494d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.546ex; height:2.176ex;" alt="{\displaystyle x_{n}\to x_{0}}" loading="lazy"> but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{n})\not \to f(x_{0})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x_{n})\not \to f(x_{0})}</annotation>

</semantics>

</math><img src="./e5be058dfe986a81975aa48ed84124b4bbe55b0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.722ex; height:2.843ex;" alt="{\displaystyle f(x_{n})\not \to f(x_{0})}" loading="lazy">, which contradicts the hypothesis of sequential continuity. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \blacksquare }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \blacksquare }</annotation>

</semantics>

</math><img src="./8733090f2d787d03101c3e16dc3f6404f0e7dd4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \blacksquare }" loading="lazy">

</td></tr></tbody></table></div>

<div class="mw-heading mw-heading4"><h4 id="Closure_operator_and_interior_operator_definitions">Closure operator and interior operator definitions</h4></div>

In terms of the <a href="Interior_(topology)" title="Interior (topology)">interior</a> and <a href="Closure_(topology)" title="Closure (topology)">closure</a> operators, we have the following equivalences,

Theorem—Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> be a mapping between topological spaces. Then the following are equivalent.

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is continuous;</li><li>for every subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subseteq Y,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\subseteq Y,}</annotation>

</semantics>

</math><img src="./8a69adc349545e2a20f31eb474960db56ccd737b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.283ex; height:2.509ex;" alt="{\displaystyle B\subseteq Y,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}\left(\operatorname {int} _{Y}B\right)\subseteq \operatorname {int} _{X}\left(f^{-1}(B)\right);}">

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}\left(\operatorname {int} _{Y}B\right)\subseteq \operatorname {int} _{X}\left(f^{-1}(B)\right);}</annotation>

</semantics>

</math><img src="./5674ae44b5cf4e385ab8f0f724bb5a57c9e59e3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.909ex; height:3.343ex;" alt="{\displaystyle f^{-1}\left(\operatorname {int} _{Y}B\right)\subseteq \operatorname {int} _{X}\left(f^{-1}(B)\right);}" loading="lazy"></li><li>for every subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\subseteq X,}</annotation>

</semantics>

</math><img src="./e8a030c36801bc01950928058517e7a7da7eaa08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.468ex; height:2.509ex;" alt="{\displaystyle A\subseteq X,}" loading="lazy">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\left(\operatorname {cl} _{X}A\right)\subseteq \operatorname {cl} _{Y}\left(f(A)\right).}</annotation>

</semantics>

</math><img src="./1df8d0524ab8fa15fe1b899b88ca89c427013131.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.468ex; height:2.843ex;" alt="{\displaystyle f\left(\operatorname {cl} _{X}A\right)\subseteq \operatorname {cl} _{Y}\left(f(A)\right).}" loading="lazy"></li></ol></div>

</div>

<th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:left;"><div style="font-size:115%;">Proof</div>

</th></tr>

<tr>

Proof. i ⇒ ii.

Fix a subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y.}</annotation>

</semantics>

</math><img src="./0c668649af47a30006f93c9847d61fee8d9ffb61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" loading="lazy"> Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {int} _{Y}B}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {int} _{Y}B}</annotation>

</semantics>

</math><img src="./eadb228dd9469b9264b01a6eff0ef99b25290c4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.481ex; height:2.509ex;" alt="{\displaystyle \operatorname {int} _{Y}B}" loading="lazy"> is open.

and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is continuous, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(\operatorname {int} _{Y}B)}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(\operatorname {int} _{Y}B)}</annotation>

</semantics>

</math><img src="./efc4368489ef215d0c11abbcd51d6b09813edee1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.944ex; height:3.176ex;" alt="{\displaystyle f^{-1}(\operatorname {int} _{Y}B)}" loading="lazy"> is open in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy">

As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {int} _{Y}B\subseteq B,}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {int} _{Y}B\subseteq B,}</annotation>

</semantics>

</math><img src="./bd304d2f3b3e1e341bb083a59f39355f49fdd008.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.991ex; height:2.509ex;" alt="{\displaystyle \operatorname {int} _{Y}B\subseteq B,}" loading="lazy"> we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(\operatorname {int} _{Y}B)\subseteq f^{-1}(B).}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(\operatorname {int} _{Y}B)\subseteq f^{-1}(B).}</annotation>

</semantics>

</math><img src="./13e6a2da0b86da047582b72595a85860f289da0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.916ex; height:3.176ex;" alt="{\displaystyle f^{-1}(\operatorname {int} _{Y}B)\subseteq f^{-1}(B).}" loading="lazy">

By the definition of the interior, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {int} _{X}\left(f^{-1}(B)\right)}">

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {int} _{X}\left(f^{-1}(B)\right)}</annotation>

</semantics>

</math><img src="./b174f87a34827be5a84ffa1341dbc162fc9cdf9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.833ex; height:3.343ex;" alt="{\displaystyle \operatorname {int} _{X}\left(f^{-1}(B)\right)}" loading="lazy"> is the largest open set contained in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(B).}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(B).}</annotation>

</semantics>

</math><img src="./67f14656f1dfd20521691751fa3e4c25dd9875f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.873ex; height:3.176ex;" alt="{\displaystyle f^{-1}(B).}" loading="lazy"> Hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(\operatorname {int} _{Y}B)\subseteq \operatorname {int} _{X}\left(f^{-1}(B)\right).}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(\operatorname {int} _{Y}B)\subseteq \operatorname {int} _{X}\left(f^{-1}(B)\right).}</annotation>

</semantics>

</math><img src="./1d9a7ef6ff3e4fdcb3fc93f60548b64f8a5e5f30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.522ex; height:3.343ex;" alt="{\displaystyle f^{-1}(\operatorname {int} _{Y}B)\subseteq \operatorname {int} _{X}\left(f^{-1}(B)\right).}" loading="lazy">

ii ⇒ iii.

Fix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\subseteq X}</annotation>

</semantics>

</math><img src="./1dce86da0107830a9a97287f9486d9b4ff022875.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.822ex; height:2.343ex;" alt="{\displaystyle A\subseteq X}" loading="lazy"> and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \operatorname {cl} _{X}A.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in \operatorname {cl} _{X}A.}</annotation>

</semantics>

</math><img src="./bb1b2b6b174a847cb4ee6104ceabd1c032bda188.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.259ex; height:2.509ex;" alt="{\displaystyle x\in \operatorname {cl} _{X}A.}" loading="lazy"> Suppose to the contrary that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\notin \operatorname {cl} _{Y}\left(f(A)\right),}">

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)\notin \operatorname {cl} _{Y}\left(f(A)\right),}</annotation>

</semantics>

</math><img src="./2470f0b1391b1dc0efe6ad881c7a7dfc0b4274eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.711ex; height:2.843ex;" alt="{\displaystyle f(x)\notin \operatorname {cl} _{Y}\left(f(A)\right),}" loading="lazy">

then we may find some open neighbourhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> that is disjoint from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{Y}\left(f(A)\right)}">

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{Y}\left(f(A)\right)}</annotation>

</semantics>

</math><img src="./0f0718d4fbea801ebf4bd2c1cac6ea6e33cc7502.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.806ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{Y}\left(f(A)\right)}" loading="lazy">. By ii, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(V)=f^{-1}(\operatorname {int} _{Y}V)\subseteq \operatorname {int} _{X}\left(f^{-1}(V)\right),}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(V)=f^{-1}(\operatorname {int} _{Y}V)\subseteq \operatorname {int} _{X}\left(f^{-1}(V)\right),}</annotation>

</semantics>

</math><img src="./78e6b260b702e50d1e2bfef0c8c700d595999301.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.917ex; height:3.343ex;" alt="{\displaystyle f^{-1}(V)=f^{-1}(\operatorname {int} _{Y}V)\subseteq \operatorname {int} _{X}\left(f^{-1}(V)\right),}" loading="lazy"> hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(V)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(V)}</annotation>

</semantics>

</math><img src="./8b5171cc2674480baf68e55ff20378a5413d4728.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.25ex; height:3.176ex;" alt="{\displaystyle f^{-1}(V)}" loading="lazy"> is open. Then we have found an open neighbourhood of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> that does not intersect <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}A}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}A}</annotation>

</semantics>

</math><img src="./76cfb33fb64dc1a062d768e91dd4d5bd3a8f07a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.442ex; height:2.509ex;" alt="{\displaystyle \operatorname {cl} _{X}A}" loading="lazy">, contradicting the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \operatorname {cl} _{X}A.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in \operatorname {cl} _{X}A.}</annotation>

</semantics>

</math><img src="./bb1b2b6b174a847cb4ee6104ceabd1c032bda188.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.259ex; height:2.509ex;" alt="{\displaystyle x\in \operatorname {cl} _{X}A.}" loading="lazy">

Hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(\operatorname {cl} _{X}A\right)\subseteq \operatorname {cl} _{Y}\left(f(A)\right).}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\left(\operatorname {cl} _{X}A\right)\subseteq \operatorname {cl} _{Y}\left(f(A)\right).}</annotation>

</semantics>

</math><img src="./1df8d0524ab8fa15fe1b899b88ca89c427013131.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.468ex; height:2.843ex;" alt="{\displaystyle f\left(\operatorname {cl} _{X}A\right)\subseteq \operatorname {cl} _{Y}\left(f(A)\right).}" loading="lazy">

iii ⇒ i.

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\subseteq Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\subseteq Y}</annotation>

</semantics>

</math><img src="./5f11391a102edf90a792fd50c970116c5f42bce7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.935ex; height:2.343ex;" alt="{\displaystyle N\subseteq Y}" loading="lazy"> be closed. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=f^{-1}(N)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M=f^{-1}(N)}</annotation>

</semantics>

</math><img src="./1e81ca5d366d76d9e9a7060ff4a2be12d8c0e576.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.067ex; height:3.176ex;" alt="{\displaystyle M=f^{-1}(N)}" loading="lazy"> be the preimage of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N.}</annotation>

</semantics>

</math><img src="./356b8b60a047de347b447f2bdafaaccf47502031.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.71ex; height:2.176ex;" alt="{\displaystyle N.}" loading="lazy">

By iii, we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(\operatorname {cl} _{X}M\right)\subseteq \operatorname {cl} _{Y}\left(f(M)\right).}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\left(\operatorname {cl} _{X}M\right)\subseteq \operatorname {cl} _{Y}\left(f(M)\right).}</annotation>

</semantics>

</math><img src="./c5001feab977636fe177781c4fc86506be82c094.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.866ex; height:2.843ex;" alt="{\displaystyle f\left(\operatorname {cl} _{X}M\right)\subseteq \operatorname {cl} _{Y}\left(f(M)\right).}" loading="lazy">

Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(M)=f(f^{-1}(N))\subseteq N,}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(M)=f(f^{-1}(N))\subseteq N,}</annotation>

</semantics>

</math><img src="./eba29c9fe1b30443e8e8ad6823322a97e005e7da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.052ex; height:3.176ex;" alt="{\displaystyle f(M)=f(f^{-1}(N))\subseteq N,}" loading="lazy">

we have further that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(\operatorname {cl} _{X}M\right)\subseteq \operatorname {cl} _{Y}N=N.}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\left(\operatorname {cl} _{X}M\right)\subseteq \operatorname {cl} _{Y}N=N.}</annotation>

</semantics>

</math><img src="./45e87d47c6045501d68305fc5eac0235d851e710.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.139ex; height:2.843ex;" alt="{\displaystyle f\left(\operatorname {cl} _{X}M\right)\subseteq \operatorname {cl} _{Y}N=N.}" loading="lazy">

Thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}M\subseteq f^{-1}\left(f(\operatorname {cl} _{X}M)\right)\subseteq f^{-1}(N)=M.}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}M\subseteq f^{-1}\left(f(\operatorname {cl} _{X}M)\right)\subseteq f^{-1}(N)=M.}</annotation>

</semantics>

</math><img src="./ae5ef7b0e5a2bd77c6a174e54985f818eff168a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.13ex; height:3.176ex;" alt="{\displaystyle \operatorname {cl} _{X}M\subseteq f^{-1}\left(f(\operatorname {cl} _{X}M)\right)\subseteq f^{-1}(N)=M.}" loading="lazy">

Hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is closed and we are done.

</td></tr></tbody></table></div>

If we declare that a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is close to a subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\subseteq X}</annotation>

</semantics>

</math><img src="./1dce86da0107830a9a97287f9486d9b4ff022875.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.822ex; height:2.343ex;" alt="{\displaystyle A\subseteq X}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \operatorname {cl} _{X}A,}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in \operatorname {cl} _{X}A,}</annotation>

</semantics>

</math><img src="./c2f3d6ab421c003fffd7ad7eef6c18cf359b8c7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.259ex; height:2.509ex;" alt="{\displaystyle x\in \operatorname {cl} _{X}A,}" loading="lazy"> then this terminology allows for a <a href="Plain_English" title="Plain English">plain English</a> description of continuity: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is continuous if and only if for every subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\subseteq X,}</annotation>

</semantics>

</math><img src="./e8a030c36801bc01950928058517e7a7da7eaa08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.468ex; height:2.509ex;" alt="{\displaystyle A\subseteq X,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> maps points that are close to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> to points that are close to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(A).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(A).}</annotation>

</semantics>

</math><img src="./43483089a73dddf063590aee1b1fc95df748c3ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.478ex; height:2.843ex;" alt="{\displaystyle f(A).}" loading="lazy"> Similarly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is continuous at a fixed given point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> if and only if whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is close to a subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\subseteq X,}</annotation>

</semantics>

</math><img src="./e8a030c36801bc01950928058517e7a7da7eaa08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.468ex; height:2.509ex;" alt="{\displaystyle A\subseteq X,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> is close to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(A).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(A).}</annotation>

</semantics>

</math><img src="./43483089a73dddf063590aee1b1fc95df748c3ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.478ex; height:2.843ex;" alt="{\displaystyle f(A).}" loading="lazy">

Instead of specifying topological spaces by their <a href="Open_set" title="Open set">open subsets</a>, any topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> can <a href="Equivalence_of_categories" title="Equivalence of categories">alternatively be determined</a> by a <a href="Kuratowski_closure_operator" class="mw-redirect" title="Kuratowski closure operator">closure operator</a> or by an <a href="Interior_operator" class="mw-redirect" title="Interior operator">interior operator</a>.

Specifically, the map that sends a subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> of a topological space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> to its <a href="Closure_(topology)" title="Closure (topology)">topological closure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}A}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}A}</annotation>

</semantics>

</math><img src="./76cfb33fb64dc1a062d768e91dd4d5bd3a8f07a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.442ex; height:2.509ex;" alt="{\displaystyle \operatorname {cl} _{X}A}" loading="lazy"> satisfies the <a href="Kuratowski_closure_axioms" title="Kuratowski closure axioms">Kuratowski closure axioms</a>. Conversely, for any <a href="Kuratowski_closure_operator" class="mw-redirect" title="Kuratowski closure operator">closure operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mapsto \operatorname {cl} A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\mapsto \operatorname {cl} A}</annotation>

</semantics>

</math><img src="./11873f8c9c5f19abc5ef73e95520c18107407ffa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.167ex; height:2.176ex;" alt="{\displaystyle A\mapsto \operatorname {cl} A}" loading="lazy"> there exists a unique topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

</semantics>

</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> (specifically, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau :=\{X\setminus \operatorname {cl} A:A\subseteq X\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau :=\{X\setminus \operatorname {cl} A:A\subseteq X\}}</annotation>

</semantics>

</math><img src="./7fd40023ea7b6edc0d56ffb028af73b4694f73be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.015ex; height:2.843ex;" alt="{\displaystyle \tau :=\{X\setminus \operatorname {cl} A:A\subseteq X\}}" loading="lazy">) such that for every subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\subseteq X,}</annotation>

</semantics>

</math><img src="./e8a030c36801bc01950928058517e7a7da7eaa08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.468ex; height:2.509ex;" alt="{\displaystyle A\subseteq X,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} A}</annotation>

</semantics>

</math><img src="./237ea361d8b480fe2b24d52dfe9e8fe8849a568f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.809ex; height:2.176ex;" alt="{\displaystyle \operatorname {cl} A}" loading="lazy"> is equal to the topological closure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{(X,\tau )}A}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{(X,\tau )}A}</annotation>

</semantics>

</math><img src="./4eecc1413f6f7f5d24494efdcc2586e5584dbb22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.028ex; height:3.009ex;" alt="{\displaystyle \operatorname {cl} _{(X,\tau )}A}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau ).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X,\tau ).}</annotation>

</semantics>

</math><img src="./f9b42cadba3d621e09a00a4e133d83ad19f6d548.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.672ex; height:2.843ex;" alt="{\displaystyle (X,\tau ).}" loading="lazy"> If the sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> are each associated with closure operators (both denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} }</annotation>

</semantics>

</math><img src="./22e35567600f75b8ce18e5cacbd15a57b97beb61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.679ex; height:2.176ex;" alt="{\displaystyle \operatorname {cl} }" loading="lazy">) then a map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> is continuous if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\operatorname {cl} A)\subseteq \operatorname {cl} (f(A))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\operatorname {cl} A)\subseteq \operatorname {cl} (f(A))}</annotation>

</semantics>

</math><img src="./c3aaa13dc7f9d2f6df700fe5f558e81052ead91b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.315ex; height:2.843ex;" alt="{\displaystyle f(\operatorname {cl} A)\subseteq \operatorname {cl} (f(A))}" loading="lazy"> for every subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\subseteq X.}</annotation>

</semantics>

</math><img src="./fd93a155fefa48b8bbb8f8a28437567f97d289eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.468ex; height:2.343ex;" alt="{\displaystyle A\subseteq X.}" loading="lazy">

Similarly, the map that sends a subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> to its <a href="Interior_(topology)" title="Interior (topology)">topological interior</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {int} _{X}A}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {int} _{X}A}</annotation>

</semantics>

</math><img src="./c36ec1da6292ef0db27136b5cb118465ba328819.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.607ex; height:2.509ex;" alt="{\displaystyle \operatorname {int} _{X}A}" loading="lazy"> defines an <a href="Interior_operator" class="mw-redirect" title="Interior operator">interior operator</a>. Conversely, any interior operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mapsto \operatorname {int} A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\mapsto \operatorname {int} A}</annotation>

</semantics>

</math><img src="./0fcc4fe40774931c3b79c1e39fd25a00d61b15d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.331ex; height:2.176ex;" alt="{\displaystyle A\mapsto \operatorname {int} A}" loading="lazy"> induces a unique topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

</semantics>

</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> (specifically, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau :=\{\operatorname {int} A:A\subseteq X\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau :=\{\operatorname {int} A:A\subseteq X\}}</annotation>

</semantics>

</math><img src="./9168d578b22aafbacd471fd2cd2e9ecd5b70227e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.005ex; height:2.843ex;" alt="{\displaystyle \tau :=\{\operatorname {int} A:A\subseteq X\}}" loading="lazy">) such that for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\subseteq X,}</annotation>

</semantics>

</math><img src="./e8a030c36801bc01950928058517e7a7da7eaa08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.468ex; height:2.509ex;" alt="{\displaystyle A\subseteq X,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {int} A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {int} A}</annotation>

</semantics>

</math><img src="./372b6f29168318a6a483acec78aefb6f82630998.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.974ex; height:2.176ex;" alt="{\displaystyle \operatorname {int} A}" loading="lazy"> is equal to the topological interior <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {int} _{(X,\tau )}A}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {int} _{(X,\tau )}A}</annotation>

</semantics>

</math><img src="./6d9ec1fb34cbeed298e143ec81969e9c42f7234e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.193ex; height:3.009ex;" alt="{\displaystyle \operatorname {int} _{(X,\tau )}A}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau ).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X,\tau ).}</annotation>

</semantics>

</math><img src="./f9b42cadba3d621e09a00a4e133d83ad19f6d548.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.672ex; height:2.843ex;" alt="{\displaystyle (X,\tau ).}" loading="lazy"> If the sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> are each associated with interior operators (both denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {int} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {int} }</annotation>

</semantics>

</math><img src="./d1223f05a56565e2dbe1688d596648f0a6579ef6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.844ex; height:2.176ex;" alt="{\displaystyle \operatorname {int} }" loading="lazy">) then a map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> is continuous if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(\operatorname {int} B)\subseteq \operatorname {int} \left(f^{-1}(B)\right)}">

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(\operatorname {int} B)\subseteq \operatorname {int} \left(f^{-1}(B)\right)}</annotation>

</semantics>

</math><img src="./2c28488e9935c4577f01205bba0f0a9bd42261cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.757ex; height:3.343ex;" alt="{\displaystyle f^{-1}(\operatorname {int} B)\subseteq \operatorname {int} \left(f^{-1}(B)\right)}" loading="lazy"> for every subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subseteq Y.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\subseteq Y.}</annotation>

</semantics>

</math><img src="./a6e0f3ddd21c78bb7c72d162c69420eadcb22c22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.283ex; height:2.343ex;" alt="{\displaystyle B\subseteq Y.}" loading="lazy"><a href="#cite_note-18">[18]</a>

<div class="mw-heading mw-heading4"><h4 id="Filters_and_prefilters">Filters and prefilters</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Filters_in_topology" title="Filters in topology">Filters in topology</a></div>

Continuity can also be characterized in terms of <a href="Filter_(set_theory)" title="Filter (set theory)">filters</a>. A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> is continuous if and only if whenever a filter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation>

</semantics>

</math><img src="./e5622de88a69f68340f8dcb43d0b8bd443ba9e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> <a href="Convergent_filter" class="mw-redirect" title="Convergent filter">converges</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> to a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation>

</semantics>

</math><img src="./e3ebdb0a09f0721ccdd0b779e0a21caf386be82a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" loading="lazy"> then the <a href="Prefilter" class="mw-redirect" title="Prefilter">prefilter</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f({\mathcal {B}})}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f({\mathcal {B}})}</annotation>

</semantics>

</math><img src="./e84b72995ed26a739921e6b18815cf5e8c06b8e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.631ex; height:2.843ex;" alt="{\displaystyle f({\mathcal {B}})}" loading="lazy"> converges in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x).}</annotation>

</semantics>

</math><img src="./e889a7f3ee0216318cbf9f3478208fa1404cc51c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.064ex; height:2.843ex;" alt="{\displaystyle f(x).}" loading="lazy"> This characterization remains true if the word "filter" is replaced by "prefilter."<a href="#cite_note-FOOTNOTEDugundji1966211–221-16">[16]</a>

<div class="mw-heading mw-heading3"><h3 id="Properties_2">Properties</h3></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:Y\to Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g:Y\to Z}</annotation>

</semantics>

</math><img src="./44e7e6d2116baaa7aa88d1adbf796ade5997a237.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.121ex; height:2.509ex;" alt="{\displaystyle g:Y\to Z}" loading="lazy"> are continuous, then so is the composition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f:X\to Z.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\circ f:X\to Z.}</annotation>

</semantics>

</math><img src="./106dee67047ba4cd8e2306a7a35adf760fd58901.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.448ex; height:2.509ex;" alt="{\displaystyle g\circ f:X\to Z.}" loading="lazy"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> is continuous and

<ul><li>X is <a href="Compact_space" title="Compact space">compact</a>, then f(X) is compact.</li>

<li>X is <a href="Connected_space" title="Connected space">connected</a>, then f(X) is connected.</li>

<li>X is <a href="Path-connected" class="mw-redirect" title="Path-connected">path-connected</a>, then f(X) is path-connected.</li>

<li>X is <a href="Lindel%C3%B6f_space" title="Lindelöf space">Lindelöf</a>, then f(X) is Lindelöf.</li>

<li>X is <a href="Separable_space" title="Separable space">separable</a>, then f(X) is separable.</li></ul>

The possible topologies on a fixed set X are <a href="Partial_ordering" class="mw-redirect" title="Partial ordering">partially ordered</a>: a topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{1}}</annotation>

</semantics>

</math><img src="./ef3e70085bbfec0d2ed0f898618bbb48db4ab7b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.07ex; height:2.009ex;" alt="{\displaystyle \tau _{1}}" loading="lazy"> is said to be <a href="Comparison_of_topologies" title="Comparison of topologies">coarser</a> than another topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{2}}</annotation>

</semantics>

</math><img src="./236a07f6109f6e8c7dd3d9229ab240a983045cd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.07ex; height:2.009ex;" alt="{\displaystyle \tau _{2}}" loading="lazy"> (notation: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{1}\subseteq \tau _{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{1}\subseteq \tau _{2}}</annotation>

</semantics>

</math><img src="./9c3324a2b43bda5848a2817396cabe4f2643ef70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.239ex; height:2.343ex;" alt="{\displaystyle \tau _{1}\subseteq \tau _{2}}" loading="lazy">) if every open subset with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{1}}</annotation>

</semantics>

</math><img src="./ef3e70085bbfec0d2ed0f898618bbb48db4ab7b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.07ex; height:2.009ex;" alt="{\displaystyle \tau _{1}}" loading="lazy"> is also open with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{2}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{2}.}</annotation>

</semantics>

</math><img src="./5bcee57d9e421b3cda35d0a37a5f849a471a0a2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.717ex; height:2.009ex;" alt="{\displaystyle \tau _{2}.}" loading="lazy"> Then, the <a href="Identity_function" title="Identity function">identity map</a>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {id} _{X}:\left(X,\tau _{2}\right)\to \left(X,\tau _{1}\right)}</annotation>

</semantics>

</math>

is continuous if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{1}\subseteq \tau _{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{1}\subseteq \tau _{2}}</annotation>

</semantics>

</math><img src="./9c3324a2b43bda5848a2817396cabe4f2643ef70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.239ex; height:2.343ex;" alt="{\displaystyle \tau _{1}\subseteq \tau _{2}}" loading="lazy"> (see also <a href="Comparison_of_topologies" title="Comparison of topologies">comparison of topologies</a>). More generally, a continuous function

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(X,\tau _{X}\right)\to \left(Y,\tau _{Y}\right)}</annotation>

</semantics>

</math>

stays continuous if the topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{Y}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{Y}}</annotation>

</semantics>

</math><img src="./ce27bc0246b4bd47ab126ac87eeb3db349978ff1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.502ex; height:2.009ex;" alt="{\displaystyle \tau _{Y}}" loading="lazy"> is replaced by a <a href="Comparison_of_topologies" title="Comparison of topologies">coarser topology</a> and/or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{X}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{X}}</annotation>

</semantics>

</math><img src="./23d277a9245384774a4f219da7d96d521a3f8240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.648ex; height:2.009ex;" alt="{\displaystyle \tau _{X}}" loading="lazy"> is replaced by a <a href="Comparison_of_topologies" title="Comparison of topologies">finer topology</a>.

<div class="mw-heading mw-heading3"><h3 id="Homeomorphisms">Homeomorphisms</h3></div>

Symmetric to the concept of a continuous map is an <a href="Open_map" class="mw-redirect" title="Open map">open map</a>, for which images of open sets are open. If an open map f has an <a href="Inverse_function" title="Inverse function">inverse function</a>, that inverse is continuous, and if a continuous map g has an inverse, that inverse is open. Given a <a href="Bijective" class="mw-redirect" title="Bijective">bijective</a> function f between two topological spaces, the inverse function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation>

</semantics>

</math><img src="./3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}" loading="lazy"> need not be continuous. A bijective continuous function with a continuous inverse function is called a <a href="Homeomorphism" title="Homeomorphism">homeomorphism</a>.

If a continuous bijection has as its <a href="Domain_of_a_function" title="Domain of a function">domain</a> a <a href="Compact_space" title="Compact space">compact space</a> and its codomain is <a href="Hausdorff_space" title="Hausdorff space">Hausdorff</a>, then it is a homeomorphism.

<div class="mw-heading mw-heading3"><h3 id="Defining_topologies_via_continuous_functions">Defining topologies via continuous functions</h3></div>

Given a function

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to S,}</annotation>

</semantics>

</math>

where X is a topological space and S is a set (without a specified topology), the <a href="Final_topology" title="Final topology">final topology</a> on S is defined by letting the open sets of S be those subsets A of S for which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(A)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(A)}</annotation>

</semantics>

</math><img src="./7b636827df9e757c54e8fe4a2067fbcb6651d31f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.206ex; height:3.176ex;" alt="{\displaystyle f^{-1}(A)}" loading="lazy"> is open in X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is <a href="Comparison_of_topologies" title="Comparison of topologies">coarser</a> than the final topology on S. Thus, the final topology is the finest topology on S that makes f continuous. If f is <a href="Surjective" class="mw-redirect" title="Surjective">surjective</a>, this topology is canonically identified with the <a href="Quotient_topology" class="mw-redirect" title="Quotient topology">quotient topology</a> under the <a href="Equivalence_relation" title="Equivalence relation">equivalence relation</a> defined by f.

Dually, for a function f from a set S to a topological space X, the <a href="Initial_topology" title="Initial topology">initial topology</a> on S is defined by designating as an open set every subset A of S such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=f^{-1}(U)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=f^{-1}(U)}</annotation>

</semantics>

</math><img src="./b14effa35e983b5740e6f40bfeb6d4ab55ca92ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.087ex; height:3.176ex;" alt="{\displaystyle A=f^{-1}(U)}" loading="lazy"> for some open subset U of X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on S. Thus, the initial topology is the coarsest topology on S that makes f continuous. If f is injective, this topology is canonically identified with the <a href="Subspace_topology" title="Subspace topology">subspace topology</a> of S, viewed as a subset of X.

A topology on a set S is uniquely determined by the class of all continuous functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\to X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\to X}</annotation>

</semantics>

</math><img src="./5566629251250ba644683a256f3ae6b6ec516d8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.093ex; height:2.176ex;" alt="{\displaystyle S\to X}" loading="lazy"> into all topological spaces X. <a href="Duality_(mathematics)" title="Duality (mathematics)">Dually</a>, a similar idea can be applied to maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to S.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\to S.}</annotation>

</semantics>

</math><img src="./147f42cb2784f666177ba382ac98a8915450690f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.74ex; height:2.176ex;" alt="{\displaystyle X\to S.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Related_notions">Related notions</h2></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:S\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:S\to Y}</annotation>

</semantics>

</math><img src="./d3934b167cb8c46ba651326059aed50ddfd4d37a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.102ex; height:2.509ex;" alt="{\displaystyle f:S\to Y}" loading="lazy"> is a continuous function from some subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> of a topological space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> then a <style data-mw-deduplicate="TemplateStyles:r1238216509">

/* start https://en.wikipedia.org/ */

.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}

/* end https://en.wikipedia.org/ */

</style>continuous extension of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is any continuous function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F:X\to Y}</annotation>

</semantics>

</math><img src="./b81de3fd0af85cab5904d269f0e3f02088cea565.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.045ex; height:2.176ex;" alt="{\displaystyle F:X\to Y}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s)=f(s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(s)=f(s)}</annotation>

</semantics>

</math><img src="./142eeec95286e4f13223ba5bc4cda26c5c5043ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.917ex; height:2.843ex;" alt="{\displaystyle F(s)=f(s)}" loading="lazy"> for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in S,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\in S,}</annotation>

</semantics>

</math><img src="./9ad060d84dec285d2822233a418fbf98689f6ddb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.077ex; height:2.509ex;" alt="{\displaystyle s\in S,}" loading="lazy"> which is a condition that often written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=F{\big \vert }_{S}.}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f=F{\big \vert }_{S}.}</annotation>

</semantics>

</math><img src="./05c33f3743188245c0c29b48b289a8f0ab703e4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.704ex; height:3.343ex;" alt="{\displaystyle f=F{\big \vert }_{S}.}" loading="lazy"> In words, it is any continuous function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F:X\to Y}</annotation>

</semantics>

</math><img src="./b81de3fd0af85cab5904d269f0e3f02088cea565.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.045ex; height:2.176ex;" alt="{\displaystyle F:X\to Y}" loading="lazy"> that <a href="Restriction_of_a_function" class="mw-redirect" title="Restriction of a function">restricts</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S.}</annotation>

</semantics>

</math><img src="./23bbb1f0f6ebdfa78b4fed06049640f7386bb44b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.146ex; height:2.176ex;" alt="{\displaystyle S.}" loading="lazy"> This notion is used, for example, in the <a href="Tietze_extension_theorem" title="Tietze extension theorem">Tietze extension theorem</a> and the <a href="Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach theorem</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:S\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:S\to Y}</annotation>

</semantics>

</math><img src="./d3934b167cb8c46ba651326059aed50ddfd4d37a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.102ex; height:2.509ex;" alt="{\displaystyle f:S\to Y}" loading="lazy"> is not continuous, then it could not possibly have a continuous extension. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is a <a href="Hausdorff_space" title="Hausdorff space">Hausdorff space</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is a <a href="Dense_set" title="Dense set">dense subset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> then a continuous extension of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:S\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:S\to Y}</annotation>

</semantics>

</math><img src="./d3934b167cb8c46ba651326059aed50ddfd4d37a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.102ex; height:2.509ex;" alt="{\displaystyle f:S\to Y}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> if one exists, will be unique. The <a href="Blumberg_theorem" title="Blumberg theorem">Blumberg theorem</a> states that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \to \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \to \mathbb {R} }</annotation>

</semantics>

</math><img src="./1e3a10a3ad05781f5cf9c2d875a02227e21a8448.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {R} }" loading="lazy"> is an arbitrary function then there exists a dense subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> such that the restriction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f{\big \vert }_{D}:D\to \mathbb {R} }">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f{\big \vert }_{D}:D\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./3231d7d3f68d407a219208189418df558029ba20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.672ex; height:3.343ex;" alt="{\displaystyle f{\big \vert }_{D}:D\to \mathbb {R} }" loading="lazy"> is continuous; in other words, every function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \to \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \to \mathbb {R} }</annotation>

</semantics>

</math><img src="./fadf3927293c3bb66c6bb34668923799af3484b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.97ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} \to \mathbb {R} }" loading="lazy"> can be restricted to some dense subset on which it is continuous.

Various other mathematical domains use the concept of continuity in different but related meanings. For example, in <a href="Order_theory" title="Order theory">order theory</a>, an order-preserving function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> between particular types of <a href="Partially_ordered_set" title="Partially ordered set">partially ordered sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is continuous if for each <a href="Directed_set" title="Directed set">directed subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup f(A)=f(\sup A).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sup f(A)=f(\sup A).}</annotation>

</semantics>

</math><img src="./0e03894b2a27b88563f23f257d47c3bdf1028164.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.184ex; height:2.843ex;" alt="{\displaystyle \sup f(A)=f(\sup A).}" loading="lazy"> Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\sup \,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,\sup \,}</annotation>

</semantics>

</math><img src="./fbcede6089537721672cb079692e5dddaf705f2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.275ex; height:2.009ex;" alt="{\displaystyle \,\sup \,}" loading="lazy"> is the <a href="Supremum" class="mw-redirect" title="Supremum">supremum</a> with respect to the orderings in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y,}">

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<annotation encoding="application/x-tex">{\displaystyle Y,}</annotation>

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</math><img src="./a3765557b7effa1a5f2f4dce9c80a25973b7009f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.42ex; height:2.509ex;" alt="{\displaystyle Y,}" loading="lazy"> respectively. This notion of continuity is the same as topological continuity when the partially ordered sets are given the <a href="Scott_topology" class="mw-redirect" title="Scott topology">Scott topology</a>.<a href="#cite_note-19">[19]</a><a href="#cite_note-20">[20]</a>

In <a href="Category_theory" title="Category theory">category theory</a>, a <a href="Functor" title="Functor">functor</a>

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<annotation encoding="application/x-tex">{\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}}</annotation>

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between two <a href="Category_(mathematics)" title="Category (mathematics)">categories</a> is called <a href="Continuous_functor" class="mw-redirect" title="Continuous functor">continuous</a> if it commutes with small <a href="Limit_(category_theory)" title="Limit (category theory)">limits</a>. That is to say,

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<annotation encoding="application/x-tex">{\displaystyle \varprojlim _{i\in I}F(C_{i})\cong F\left(\varprojlim _{i\in I}C_{i}\right)}</annotation>

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for any small (that is, indexed by a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I,}">

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<annotation encoding="application/x-tex">{\displaystyle I,}</annotation>

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</math><img src="./a12504e3a6f191d6fb24fb4a6795266bdd171664.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.819ex; height:2.509ex;" alt="{\displaystyle I,}" loading="lazy"> as opposed to a <a href="Class_(mathematics)" class="mw-redirect" title="Class (mathematics)">class</a>) <a href="Diagram_(category_theory)" title="Diagram (category theory)">diagram</a> of <a href="Object_(category_theory)" class="mw-redirect" title="Object (category theory)">objects</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

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</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy">.

A <a href="Continuity_space" class="mw-redirect" title="Continuity space">continuity space</a> is a generalization of metric spaces and posets,<a href="#cite_note-21">[21]</a><a href="#cite_note-22">[22]</a> which uses the concept of <a href="Quantale" title="Quantale">quantales</a>, and that can be used to unify the notions of metric spaces and <a href="Domain_theory" title="Domain theory">domains</a>.<a href="#cite_note-23">[23]</a>

In <a href="Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a>, a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:E\to \mathbb {R} ^{k}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle f:E\to \mathbb {R} ^{k}}</annotation>

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</math><img src="./3f52da1214e297cf6b9cc59b60bdea860bad7503.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.372ex; height:3.009ex;" alt="{\displaystyle f:E\to \mathbb {R} ^{k}}" loading="lazy"> defined on a <a href="Lebesgue_measurable_set" class="mw-redirect" title="Lebesgue measurable set">Lebesgue measurable set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\subseteq \mathbb {R} ^{n}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle E\subseteq \mathbb {R} ^{n}}</annotation>

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</math><img src="./8ce152637bd9dc168eb02f359868370850adf5c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.771ex; height:2.509ex;" alt="{\displaystyle E\subseteq \mathbb {R} ^{n}}" loading="lazy"> is called <a href="Approximately_continuous" class="mw-redirect" title="Approximately continuous">approximately continuous</a> at a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in E}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{0}\in E}</annotation>

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</math><img src="./96b8114a859b2795587b94eea597ac93f164eb69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7ex; height:2.509ex;" alt="{\displaystyle x_{0}\in E}" loading="lazy"> if the <a href="Approximate_limit" title="Approximate limit">approximate limit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

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</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> exists and equals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{0})}">

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f(x_{0})}</annotation>

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</math><img src="./27cf1dbaefc6038a22779fb2943aff758a592a3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.472ex; height:2.843ex;" alt="{\displaystyle f(x_{0})}" loading="lazy">. This generalizes the notion of continuity by replacing the ordinary limit with the <a href="Approximate_limit" title="Approximate limit">approximate limit</a>. A fundamental result known as the <a href="Stepanov-Denjoy_theorem" class="mw-redirect" title="Stepanov-Denjoy theorem">Stepanov-Denjoy theorem</a> states that a function is <a href="Measurable_function" title="Measurable function">measurable</a> if and only if it is approximately continuous <a href="Almost_everywhere" title="Almost everywhere">almost everywhere</a>.<a href="#cite_note-24">[24]</a>

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<ul><li><a href="Continuity_(mathematics)" class="mw-redirect" title="Continuity (mathematics)">Continuity (mathematics)</a></li>

<li><a href="Absolute_continuity" title="Absolute continuity">Absolute continuity</a></li>

<li><a href="Approximate_continuity" class="mw-redirect" title="Approximate continuity">Approximate continuity</a></li>

<li><a href="Dini_continuity" title="Dini continuity">Dini continuity</a></li>

<li><a href="Equicontinuity" title="Equicontinuity">Equicontinuity</a></li>

<li><a href="Geometric_continuity" class="mw-redirect" title="Geometric continuity">Geometric continuity</a></li>

<li><a href="Parametric_continuity" class="mw-redirect" title="Parametric continuity">Parametric continuity</a></li>

<li><a href="Classification_of_discontinuities" title="Classification of discontinuities">Classification of discontinuities</a></li>

<li><a href="Coarse_function" title="Coarse function">Coarse function</a></li>

<li><a href="Continuous_function_(set_theory)" title="Continuous function (set theory)">Continuous function (set theory)</a></li>

<li><a href="Continuous_stochastic_process" title="Continuous stochastic process">Continuous stochastic process</a></li>

<li><a href="Normal_function" title="Normal function">Normal function</a></li>

<li><a href="Open_and_closed_maps" title="Open and closed maps">Open and closed maps</a></li>

<li><a href="Piecewise" class="mw-redirect" title="Piecewise">Piecewise</a></li>

<li><a href="Symmetrically_continuous_function" title="Symmetrically continuous function">Symmetrically continuous function</a></li></ul>

</div>

<ul><li><a href="Direction-preserving_function" title="Direction-preserving function">Direction-preserving function</a> - an analog of a continuous function in discrete spaces.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Continuity_(functions)" class="extiw external" title="commons:Category:Continuity (functions)">Continuity (functions)</a>.</div></div>

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.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite id="CITEREFBolzano1817" class="citation web cs1">Bolzano, Bernard (1817). <a rel="nofollow" class="external text" href="http://dml.cz/handle/10338.dmlcz/400352">"Rein analytischer Beweis des Lehrsatzes daß zwischen je zwey Werthen, die ein entgegengesetzetes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege"</a>. Prague: Haase.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFDugac1973" class="citation cs2">Dugac, Pierre (1973), "Eléments d'Analyse de Karl Weierstrass", Archive for History of Exact Sciences, 10 (1–2): 41–176, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf00343406">10.1007/bf00343406</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122843140">122843140</a></cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFGoursat1904" class="citation cs2">Goursat, E. (1904), A course in mathematical analysis, Boston: Ginn, p. 2</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFJordan1893" class="citation cs2">Jordan, M.C. (1893), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=h2VKAAAAMAAJ&pg=PA46">Cours d'analyse de l'École polytechnique</a>, vol. 1 (2nd ed.), Paris: Gauthier-Villars, p. 46</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFHarper2016" class="citation cs2">Harper, J.F. (2016), "Defining continuity of real functions of real variables", BSHM Bulletin: Journal of the British Society for the History of Mathematics, 31 (3): 1–16, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F17498430.2015.1116053">10.1080/17498430.2015.1116053</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123997123">123997123</a></cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFRusnockKerr-Lawson2005" class="citation cs2">Rusnock, P.; Kerr-Lawson, A. (2005), "Bolzano and uniform continuity", Historia Mathematica, 32 (3): 303–311, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.hm.2004.11.003">10.1016/j.hm.2004.11.003</a></cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFStrang1991" class="citation book cs1">Strang, Gilbert (1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OisInC1zvEMC&pg=PA87">Calculus</a>. SIAM. p. 702. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0961408820</bdi>.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFSpeck2014" class="citation web cs1">Speck, Jared (2014). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161006014646/http://math.mit.edu/~jspeck/18.01_Fall%202014/Supplementary%20notes/01c.pdf">"Continuity and Discontinuity"</a> (PDF). MIT Math. p. 3. Archived from <a rel="nofollow" class="external text" href="http://math.mit.edu/~jspeck/18.01_Fall%202014/Supplementary%20notes/01c.pdf">the original</a> (PDF) on 2016-10-06. Retrieved 2016-09-02. <q>Example 5. The function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/x}">

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<annotation encoding="application/x-tex">{\displaystyle 1/x}</annotation>

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</math><img src="./a55fefc6f37f48a9b4414b09ad3b17dfa739d9e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle 1/x}" loading="lazy"> is continuous on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,\infty )}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (0,\infty )}</annotation>

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</math><img src="./da17102e4ed0886686094ee531df040d2e86352a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.329ex; height:2.843ex;" alt="{\displaystyle (0,\infty )}" loading="lazy"> and on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,0),}">

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<annotation encoding="application/x-tex">{\displaystyle (-\infty ,0),}</annotation>

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</math><img src="./08a94033cbd412d6902370e42f1bdc90acc08bd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.784ex; height:2.843ex;" alt="{\displaystyle (-\infty ,0),}" loading="lazy">, i.e., for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x>0}</annotation>

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</math><img src="./80d24be5f0eb4a9173da6038badc8659546021d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>0}" loading="lazy"> and for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<0,}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x<0,}</annotation>

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</math><img src="./878f7bcba151597923cfab9a9f13c880cbae59cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.237ex; height:2.509ex;" alt="{\displaystyle x<0,}" loading="lazy"> in other words, at every point in its domain. However, it is not a continuous function since its domain is not an interval. It has a single point of discontinuity, namely <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0,}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x=0,}</annotation>

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</math><img src="./d54be06efe9f69b9bfb720190b5f29c76944a45b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.237ex; height:2.509ex;" alt="{\displaystyle x=0,}" loading="lazy">, and an infinite discontinuity there.</q></cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFLang1997" class="citation cs2"><a href="Serge_Lang" title="Serge Lang">Lang, Serge</a> (1997), Undergraduate analysis, <a href="Undergraduate_Texts_in_Mathematics" title="Undergraduate Texts in Mathematics">Undergraduate Texts in Mathematics</a> (2nd ed.), Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-94841-6</bdi></cite>, section II.4

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <a rel="nofollow" class="external text" href="http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF">Introduction to Real Analysis</a>, updated April 2010, William F. Trench, Theorem 3.5.2, p. 172

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <a rel="nofollow" class="external text" href="http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF">Introduction to Real Analysis</a>, updated April 2010, William F. Trench, 3.5 "A More Advanced Look at the Existence of the Proper Riemann Integral", pp. 171–177

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.math.wisc.edu/~keisler/calc.html">"Elementary Calculus"</a>. wisc.edu.</cite>

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFBrown2009" class="citation cs2">Brown, James Ward (2009), Complex Variables and Applications (8th ed.), McGraw Hill, p. 54, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-305194-9</bdi></cite>

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFGaal2009" class="citation cs2">Gaal, Steven A. (2009), Point set topology, New York: <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-47222-5</bdi></cite>, section IV.10

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> <cite id="CITEREFSearcóid2006" class="citation cs2">Searcóid, Mícheál Ó (2006), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aP37I4QWFRcC">Metric spaces</a>, Springer undergraduate mathematics series, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-84628-369-7</bdi></cite>, section 9.4

</li>

<li id="cite_note-FOOTNOTEDugundji1966211–221-16">^ <a href="#cite_ref-FOOTNOTEDugundji1966211–221_16-0">a</a> <a href="#cite_ref-FOOTNOTEDugundji1966211–221_16-1">b</a> <a href="#cite_ref-FOOTNOTEDugundji1966211–221_16-2">c</a> <a href="#CITEREFDugundji1966">Dugundji 1966</a>, pp. 211–221.

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite id="CITEREFShurman2016" class="citation book cs1">Shurman, Jerry (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wTmgDQAAQBAJ">Calculus and Analysis in Euclidean Space</a> (illustrated ed.). Springer. pp. 271–272. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-49314-5</bdi>.</cite>

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/q/1209229">"general topology - Continuity and interior"</a>. Mathematics Stack Exchange.</cite>

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> <cite id="CITEREFGoubault-Larrecq2013" class="citation book cs1">Goubault-Larrecq, Jean (2013). Non-Hausdorff Topology and Domain Theory: Selected Topics in Point-Set Topology. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1107034136</bdi>.</cite>

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> <cite id="CITEREFGierzHofmannKeimelLawson2003" class="citation book cs1">Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M. W.; Scott, D. S. (2003). <a rel="nofollow" class="external text" href="https://archive.org/details/continuouslattic0000unse">Continuous Lattices and Domains</a>. Encyclopedia of Mathematics and its Applications. Vol. 93. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0521803381</bdi>.</cite>

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> <cite id="CITEREFFlagg1997" class="citation journal cs1">Flagg, R. C. (1997). "Quantales and continuity spaces". Algebra Universalis. 37 (3): 257–276. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.48.851">10.1.1.48.851</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs000120050018">10.1007/s000120050018</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17603865">17603865</a>.</cite>

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> <cite id="CITEREFKopperman1988" class="citation journal cs1">Kopperman, R. (1988). "All topologies come from generalized metrics". American Mathematical Monthly. 95 (2): 89–97. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2323060">10.2307/2323060</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2323060">2323060</a>.</cite>

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> <cite id="CITEREFFlaggKopperman1997" class="citation journal cs1">Flagg, B.; Kopperman, R. (1997). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0304-3975%2897%2900236-3">"Continuity spaces: Reconciling domains and metric spaces"</a>. Theoretical Computer Science. 177 (1): 111–138. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0304-3975%2897%2900236-3">10.1016/S0304-3975(97)00236-3</a>.</cite>

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <cite id="CITEREFFederer1969" class="citation book cs1">Federer, H. (1969). Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften. Vol. 153. New York: Springer-Verlag.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

<ul><li><cite id="CITEREFDugundji1966" class="citation book cs1"><a href="James_Dugundji" title="James Dugundji">Dugundji, James</a> (1966). Topology. Boston: Allyn and Bacon. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-697-06889-7</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/395340485">395340485</a>.</cite></li>

<li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Continuous_function">"Continuous function"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Calculus249" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Calculus249" style="font-size:114%;margin:0 4em"><a href="Calculus" title="Calculus">Calculus</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Precalculus" title="Precalculus">Precalculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Binomial_theorem" title="Binomial theorem">Binomial theorem</a></li>

<li><a href="Concave_function" title="Concave function">Concave function</a></li>

<li><a href="Factorial" title="Factorial">Factorial</a></li>

<li><a href="Finite_difference" title="Finite difference">Finite difference</a></li>

<li><a href="Free_variables_and_bound_variables" title="Free variables and bound variables">Free variables and bound variables</a></li>

<li><a href="Graph_of_a_function" title="Graph of a function">Graph of a function</a></li>

<li><a href="Linear_function" title="Linear function">Linear function</a></li>

<li><a href="Radian" title="Radian">Radian</a></li>

<li><a href="Rolle's_theorem" title="Rolle's theorem">Rolle's theorem</a></li>

<li><a href="Secant_line" title="Secant line">Secant</a></li>

<li><a href="Slope" title="Slope">Slope</a></li>

<li><a href="Tangent" title="Tangent">Tangent</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Limit_(mathematics)" title="Limit (mathematics)">Limits</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Indeterminate_form" title="Indeterminate form">Indeterminate form</a></li>

<li><a href="Limit_of_a_function" title="Limit of a function">Limit of a function</a>

<ul><li><a href="One-sided_limit" title="One-sided limit">One-sided limit</a></li></ul></li>

<li><a href="Limit_of_a_sequence" title="Limit of a sequence">Limit of a sequence</a></li>

<li><a href="Order_of_approximation" title="Order of approximation">Order of approximation</a></li>

<li><a href="(%CE%B5%2C_%CE%B4)-definition_of_limit" class="mw-redirect" title="(ε, δ)-definition of limit">(ε, δ)-definition of limit</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Differential_calculus" title="Differential calculus">Differential calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Derivative" title="Derivative">Derivative</a></li>

<li><a href="Second_derivative" title="Second derivative">Second derivative</a></li>

<li><a href="Partial_derivative" title="Partial derivative">Partial derivative</a></li>

<li><a href="Differential_(mathematics)" title="Differential (mathematics)">Differential</a></li>

<li><a href="Differential_operator" title="Differential operator">Differential operator</a></li>

<li><a href="Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li>

<li><a href="Notation_for_differentiation" title="Notation for differentiation">Notation</a>

<ul><li><a href="Leibniz's_notation" title="Leibniz's notation">Leibniz's notation</a></li>

<li><a href="Newton's_notation_for_differentiation" class="mw-redirect" title="Newton's notation for differentiation">Newton's notation</a></li></ul></li>

<li><a href="Differentiation_rules" title="Differentiation rules">Rules of differentiation</a>

<ul><li><a href="Linearity_of_differentiation" title="Linearity of differentiation">linearity</a></li>

<li><a href="Power_rule" title="Power rule">Power</a></li>

<li><a href="Chain_rule" title="Chain rule">Chain</a></li>

<li><a href="L'H%C3%B4pital's_rule" title="L'Hôpital's rule">L'Hôpital's</a></li>

<li><a href="Product_rule" title="Product rule">Product</a>

<ul><li><a href="General_Leibniz_rule" title="General Leibniz rule">General Leibniz's rule</a></li></ul></li>

<li><a href="Quotient_rule" title="Quotient rule">Quotient</a></li></ul></li>

<li>Other techniques

<ul><li><a href="Implicit_differentiation" class="mw-redirect" title="Implicit differentiation">Implicit differentiation</a></li>

<li><a href="Inverse_functions_and_differentiation" class="mw-redirect" title="Inverse functions and differentiation">Inverse functions and differentiation</a></li>

<li><a href="Logarithmic_derivative" title="Logarithmic derivative">Logarithmic derivative</a></li>

<li><a href="Related_rates" title="Related rates">Related rates</a></li></ul></li>

<li><a href="Stationary_point" title="Stationary point">Stationary points</a>

<ul><li><a href="First_derivative_test" class="mw-redirect" title="First derivative test">First derivative test</a></li>

<li><a href="Second_derivative_test" class="mw-redirect" title="Second derivative test">Second derivative test</a></li>

<li><a href="Extreme_value_theorem" title="Extreme value theorem">Extreme value theorem</a></li>

<li><a href="Maximum_and_minimum" title="Maximum and minimum">Maximum and minimum</a></li></ul></li>

<li>Further applications

<ul><li><a href="Newton's_method" title="Newton's method">Newton's method</a></li>

<li><a href="Taylor's_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></li>

<li><a href="Differential_equation" title="Differential equation">Differential equation</a>

<ul><li><a href="Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equation</a></li>

<li><a href="Partial_differential_equation" title="Partial differential equation">Partial differential equation</a></li>

<li><a href="Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Integral_calculus" class="mw-redirect" title="Integral calculus">Integral calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Antiderivative" title="Antiderivative">Antiderivative</a></li>

<li><a href="Arc_length" title="Arc length">Arc length</a></li>

<li><a href="Riemann_integral" title="Riemann integral">Riemann integral</a></li>

<li><a href="Integral#Properties" title="Integral">Basic properties</a></li>

<li><a href="Constant_of_integration" title="Constant of integration">Constant of integration</a></li>

<li><a href="Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem of calculus</a>

<ul><li><a href="Leibniz_integral_rule" title="Leibniz integral rule">Differentiating under the integral sign</a></li></ul></li>

<li><a href="Integration_by_parts" title="Integration by parts">Integration by parts</a></li>

<li><a href="Integration_by_substitution" title="Integration by substitution">Integration by substitution</a>

<ul><li><a href="Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a></li>

<li><a href="Euler_substitution" title="Euler substitution">Euler</a></li>

<li><a href="Tangent_half-angle_substitution" title="Tangent half-angle substitution">Tangent half-angle substitution</a></li></ul></li>

<li><a href="Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions in integration</a>

<ul><li><a href="Quadratic_integral" title="Quadratic integral">Quadratic integral</a></li></ul></li>

<li><a href="Trapezoidal_rule" title="Trapezoidal rule">Trapezoidal rule</a></li>

<li>Volumes

<ul><li><a href="Disc_integration" title="Disc integration">Washer method</a></li>

<li><a href="Shell_integration" title="Shell integration">Shell method</a></li></ul></li>

<li><a href="Integral_equation" title="Integral equation">Integral equation</a></li>

<li><a href="Integro-differential_equation" title="Integro-differential equation">Integro-differential equation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Vector_calculus" title="Vector calculus">Vector calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>Derivatives

<li><a href="Directional_derivative" title="Directional derivative">Directional derivative</a></li>

<li><a href="Divergence" title="Divergence">Divergence</a></li>

<li><a href="Gradient" title="Gradient">Gradient</a></li>

<li><a href="Laplace_operator" title="Laplace operator">Laplacian</a></li></ul></li>

<li>Basic theorems

<ul><li><a href="Fundamental_Theorem_of_Line_Integrals" class="mw-redirect" title="Fundamental Theorem of Line Integrals">Line integrals</a></li>

<li><a href="Green's_theorem" title="Green's theorem">Green's</a></li>

<li><a href="Stokes'_theorem" title="Stokes' theorem">Stokes'</a></li>

<li><a href="Divergence_theorem" title="Divergence theorem">Gauss'</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Divergence_theorem" title="Divergence theorem">Divergence theorem</a></li>

<li><a href="Geometric_calculus" title="Geometric calculus">Geometric</a></li>

<li><a href="Hessian_matrix" title="Hessian matrix">Hessian matrix</a></li>

<li><a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian matrix and determinant</a></li>

<li><a href="Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li>

<li><a href="Line_integral" title="Line integral">Line integral</a></li>

<li><a href="Matrix_calculus" title="Matrix calculus">Matrix</a></li>

<li><a href="Multiple_integral" title="Multiple integral">Multiple integral</a></li>

<li><a href="Partial_derivative" title="Partial derivative">Partial derivative</a></li>

<li><a href="Surface_integral" title="Surface integral">Surface integral</a></li>

<li><a href="Volume_integral" title="Volume integral">Volume integral</a></li>

<li>Advanced topics

<ul><li><a href="Differential_form" title="Differential form">Differential forms</a></li>

<li><a href="Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li>

<li><a href="Generalized_Stokes'_theorem" class="mw-redirect" title="Generalized Stokes' theorem">Generalized Stokes' theorem</a></li>

<li><a href="Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sequences and series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Arithmetico-geometric_sequence" title="Arithmetico-geometric sequence">Arithmetico-geometric sequence</a></li>

<li>Types of series

<ul><li><a href="Alternating_series" title="Alternating series">Alternating</a></li>

<li><a href="Binomial_series" title="Binomial series">Binomial</a></li>

<li><a href="Fourier_series" title="Fourier series">Fourier</a></li>

<li><a href="Geometric_series" title="Geometric series">Geometric</a></li>

<li><a href="Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li>

<li><a href="Infinite_series" class="mw-redirect" title="Infinite series">Infinite</a></li>

<li><a href="Power_series" title="Power series">Power</a>

<ul><li><a href="Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin</a></li>

<li><a href="Taylor_series" title="Taylor series">Taylor</a></li></ul></li>

<li><a href="Telescoping_series" title="Telescoping series">Telescoping</a></li></ul></li>

<li>Tests of convergence

<li><a href="Alternating_series_test" title="Alternating series test">Alternating series</a></li>

<li><a href="Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li>

<li><a href="Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li>

<li><a href="Dirichlet's_test" title="Dirichlet's test">Dirichlet's</a></li>

<li><a href="Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li>

<li><a href="Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li>

<li><a href="Ratio_test" title="Ratio test">Ratio</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special functions and numbers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bernoulli_number" title="Bernoulli number">Bernoulli numbers</a></li>

<li><a href="E_(mathematical_constant)" title="E (mathematical constant)">e (mathematical constant)</a></li>

<li><a href="Exponential_function" title="Exponential function">Exponential function</a></li>

<li><a href="Natural_logarithm" title="Natural logarithm">Natural logarithm</a></li>

<li><a href="Stirling's_approximation" title="Stirling's approximation">Stirling's approximation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="History_of_calculus" title="History of calculus">History of calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adequality" title="Adequality">Adequality</a></li>

<li><a href="Brook_Taylor" title="Brook Taylor">Brook Taylor</a></li>

<li><a href="Colin_Maclaurin" title="Colin Maclaurin">Colin Maclaurin</a></li>

<li><a href="Generality_of_algebra" title="Generality of algebra">Generality of algebra</a></li>

<li><a href="Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li>

<li><a href="Infinitesimal" title="Infinitesimal">Infinitesimal</a></li>

<li><a href="Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">Infinitesimal calculus</a></li>

<li><a href="Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li>

<li><a href="Fluxion" title="Fluxion">Fluxion</a></li>

<li><a href="Law_of_Continuity" class="mw-redirect" title="Law of Continuity">Law of Continuity</a></li>

<li><a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li>

<li><a href="Method_of_Fluxions" title="Method of Fluxions">Method of Fluxions</a></li>

<li><a href="The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Integrals32" scope="row" class="navbox-group" style="width:1%;text-align:left"><a href="Lists_of_integrals" title="Lists of integrals">Integrals</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_integrals_of_rational_functions" title="List of integrals of rational functions">rational functions</a></li>

<li><a href="List_of_integrals_of_irrational_algebraic_functions" title="List of integrals of irrational algebraic functions">irrational algebraic functions</a></li>

<li><a href="List_of_integrals_of_exponential_functions" title="List of integrals of exponential functions">exponential functions</a></li>

<li><a href="List_of_integrals_of_logarithmic_functions" title="List of integrals of logarithmic functions">logarithmic functions</a></li>

<li><a href="List_of_integrals_of_hyperbolic_functions" title="List of integrals of hyperbolic functions">hyperbolic functions</a>

<ul><li><a href="List_of_integrals_of_inverse_hyperbolic_functions" title="List of integrals of inverse hyperbolic functions">inverse</a></li></ul></li>

<li><a href="List_of_integrals_of_trigonometric_functions" title="List of integrals of trigonometric functions">trigonometric functions</a>

<ul><li><a href="List_of_integrals_of_inverse_trigonometric_functions" title="List of integrals of inverse trigonometric functions">inverse</a></li>

<li><a href="Integral_of_the_secant_function" title="Integral of the secant function">Secant</a></li>

<li><a href="Integral_of_secant_cubed" title="Integral of secant cubed">Secant cubed</a></li></ul></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_limits" title="List of limits">List of limits</a></li>

<li><a href="Differentiation_rules" title="Differentiation rules">List of derivatives</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous topics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>Complex calculus

<ul><li><a href="Contour_integral" class="mw-redirect" title="Contour integral">Contour integral</a></li></ul></li>

<li>Differential geometry

<ul><li><a href="Manifold" title="Manifold">Manifold</a></li>

<li><a href="Curvature" title="Curvature">Curvature</a></li>

<li><a href="Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">of curves</a></li>

<li><a href="Differential_geometry_of_surfaces" title="Differential geometry of surfaces">of surfaces</a></li>

<li><a href="Tensor" title="Tensor">Tensor</a></li></ul></li>

<li><a href="Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a></li>

<li><a href="Gabriel's_horn" title="Gabriel's horn">Gabriel's horn</a></li>

<li><a href="Integration_Bee" title="Integration Bee">Integration Bee</a></li>

<li><a href="Proof_that_22/7_exceeds_%CF%80" title="Proof that 22/7 exceeds π">Proof that 22/7 exceeds π</a></li>

<li><a href="Regiomontanus'_angle_maximization_problem" title="Regiomontanus' angle maximization problem">Regiomontanus' angle maximization problem</a></li>

<li><a href="Steinmetz_solid" title="Steinmetz solid">Steinmetz solid</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Major_topics_in_mathematical_analysis88" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Major_topics_in_mathematical_analysis88" style="font-size:114%;margin:0 4em">Major topics in <a href="Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Calculus" title="Calculus">Calculus</a>: <a href="Integral" title="Integral">Integration</a></li>

<li><a href="Derivative" title="Derivative">Differentiation</a></li>

<li><a href="Differential_equation" title="Differential equation">Differential equations</a>

<ul><li><a href="Ordinary_differential_equation" title="Ordinary differential equation">ordinary</a></li>

<li><a href="Partial_differential_equation" title="Partial differential equation">partial</a></li>

<li><a href="Stochastic_differential_equation" title="Stochastic differential equation">stochastic</a></li></ul></li>

<li><a href="Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem of calculus</a></li>

<li><a href="Calculus_of_variations" title="Calculus of variations">Calculus of variations</a></li>

<li><a href="Vector_calculus" title="Vector calculus">Vector calculus</a></li>

<li><a href="Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li>

<li><a href="Matrix_calculus" title="Matrix calculus">Matrix calculus</a></li>

<li><a href="Lists_of_integrals" title="Lists of integrals">Lists of integrals</a></li>

<li><a href="Table_of_derivatives" class="mw-redirect" title="Table of derivatives">Table of derivatives</a></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Real_analysis" title="Real analysis">Real analysis</a></li>

<li><a href="Complex_analysis" title="Complex analysis">Complex analysis</a></li>

<li><a href="Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a> (<a href="Quaternionic_analysis" title="Quaternionic analysis">quaternionic analysis</a>)</li>

<li><a href="Functional_analysis" title="Functional analysis">Functional analysis</a></li>

<li><a href="Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li>

<li><a href="Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li>

<li><a href="Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li>

<li><a href="P-adic_analysis" title="P-adic analysis">P-adic analysis</a> (<a href="P-adic_number" title="P-adic number">P-adic numbers</a>)</li>

<li><a href="Measure_(mathematics)" title="Measure (mathematics)">Measure theory</a></li>

<li><a href="Representation_theory" title="Representation theory">Representation theory</a></li></ul>

<ul><li><a href="Function_(mathematics)" title="Function (mathematics)">Functions</a></li>

<li><a href="Special_functions" title="Special functions">Special functions</a></li>

<li><a href="Limit_(mathematics)" title="Limit (mathematics)">Limit</a></li>

<li><a href="Series_(mathematics)" title="Series (mathematics)">Series</a></li>

<li><a href="Infinity" title="Infinity">Infinity</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Function330" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Function330" style="font-size:114%;margin:0 4em"><a href="Function_(mathematics)" title="Function (mathematics)">Function</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="History_of_the_function_concept" title="History of the function concept">History</a></li>

<li><a href="List_of_mathematical_functions" title="List of mathematical functions">List of specific functions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types by domain and codomain</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classes/properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Constant_function" title="Constant function">Constant</a></li>

<li><a href="Identity_function" title="Identity function">Identity</a></li>

<li><a href="Linear_map" title="Linear map">Linear</a></li>

<li><a href="Polynomial" title="Polynomial">Polynomial</a></li>

<li><a href="Rational_function" title="Rational function">Rational</a></li>

<li><a href="Algebraic_function" title="Algebraic function">Algebraic</a></li>

<li><a href="Analytic_function" title="Analytic function">Analytic</a></li>

<li><a href="Smooth_function" class="mw-redirect" title="Smooth function">Smooth</a></li>

<li><a href="Measurable_function" title="Measurable function">Measurable</a></li>

<li><a href="Injective_function" title="Injective function">Injective</a></li>

<li><a href="Surjective_function" title="Surjective function">Surjective</a></li>

<li><a href="Bijection" title="Bijection">Bijective</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Restriction_(mathematics)" title="Restriction (mathematics)">Restriction</a></li>

<li><a href="Function_composition" title="Function composition">Composition</a></li>

<li><a href="Inverse_function" title="Inverse function">Inverse</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Relation_(mathematics)" title="Relation (mathematics)">Relation</a> (<a href="Binary_relation" title="Binary relation">Binary relation</a>)</li>

<li><a href="Set-valued_function" title="Set-valued function">Set-valued</a></li>

<li><a href="Multivalued_function" title="Multivalued function">Multivalued</a></li>

<li><a href="Partial_function" title="Partial function">Partial</a></li>

<li><a href="Implicit_function" title="Implicit function">Implicit</a></li>

<li><a href="Function_space" title="Function space">Space</a></li>

<li><a href="Higher-order_function" title="Higher-order function">Higher-order</a></li>

<li><a href="Morphism" title="Morphism">Morphism</a></li>

<li><a href="Functor" title="Functor">Functor</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li> Category</li></ul>

</div></td></tr></tbody></table></div>

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