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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Integer-valued function</span></span>
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</style><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="letter-spacing:0.0125em; background-color:#FFCC99"><a href="Function_(mathematics)" title="Function (mathematics)">Function</a></th></tr><tr><td class="sidebar-image"><span class="texhtml texhtml-big" style="font-size:250%;"><i>x</i> ↦ <i>f</i> (<i>x</i>)</span></td></tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px">
<a href="History_of_the_function_concept" title="History of the function concept">History of the function concept</a></th></tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px">
Types by <a href="Domain_of_a_function" title="Domain of a function">domain</a> and <a href="Codomain" title="Codomain">codomain</a></th></tr><tr><td class="sidebar-content">
<div class="hlist">
<ul><li><a href="Boolean-valued_function" title="Boolean-valued function"><span class="texhtml"><span title="arbitrary set"><var>X</var></span> → <span title="Codomain of Booleans">𝔹</span></span></a></li>
<li><a href="Ordered_pair" title="Ordered pair">
<span class="texhtml"><span title="Domain of Booleans">𝔹</span>
→ <span title="arbitrary set"><var>X</var></span></span></a></li>
<li><a href="Boolean_function" title="Boolean function">
<span class="texhtml"><span title="several Boolean variables">𝔹<sup><var>n</var></sup></span>
→ <span title="Codomain of natural numbers"><var>X</var></span></span></a></li>
</ul><p>
</p>
<ul><li><a href="Sequence" title="Sequence">
<span class="texhtml"><span title="integers">ℤ</span>
→ <span title="arbitrary set"><var>X</var></span></span></a></li>
<li><a href="Real-valued_function" title="Real-valued function">
<span class="texhtml"><span title="arbitrary set"><var>X</var></span>
→ <span title="real numbers">ℝ</span></span></a></li>
<li><a href="Function_of_a_real_variable" title="Function of a real variable">
<span class="texhtml"><span title="real numbers">ℝ</span>
→ <span title="arbitrary set"><var>X</var></span></span></a></li>
<li><a href="Function_of_several_real_variables" title="Function of several real variables">
<span class="texhtml"><span title="real coordinate (or Euclidean) space">ℝ<sup><var>n</var></sup></span>
→ <span title="arbitrary set"><var>X</var></span></span></a></li>
<li><a href="Complex-valued_function" class="mw-redirect" title="Complex-valued function">
<span class="texhtml"><span title="arbitrary set"><var>X</var></span>
→ <span title="complex numbers">ℂ</span></span></a></li>
<li><a href="Function_of_a_complex_variable" class="mw-redirect" title="Function of a complex variable">
<span class="texhtml"><span title="complex numbers">ℂ</span>
→ <span title="arbitrary set"><var>X</var></span></span></a></li>
<li><a href="Function_of_several_complex_variables" title="Function of several complex variables">
<span class="texhtml"><span title="complex coordinate space">ℂ<sup><var>n</var></sup></span>
→ <span title="arbitrary set"><var>X</var></span></span></a></li></ul>
</div></td>
</tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px">
<a href="List_of_types_of_functions" title="List of types of functions">Classes/properties</a> </th></tr><tr><td class="sidebar-content">
<div class="hlist">
<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="Continuous_function" title="Continuous function">Continuous</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 class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px">
Constructions</th></tr><tr><td class="sidebar-content">
<div class="hlist">
<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="Lambda_calculus" title="Lambda calculus">λ</a></li>
<li><a href="Inverse_function" title="Inverse function">Inverse</a></li></ul>
</div></td>
</tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px">
Generalizations </th></tr><tr><td class="sidebar-content">
<div class="hlist">
<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><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px">
<a href="List_of_mathematical_functions" title="List of mathematical functions">List of specific functions</a></th></tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">
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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, an <b>integer-valued function</b> is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> whose values are <a href="Integer" title="Integer">integers</a>. In other words, it is a function that assigns an integer to each member of its <a href="Domain_of_a_function" title="Domain of a function">domain</a>.
</p><p>The <a href="Floor_and_ceiling_functions" title="Floor and ceiling functions">floor and ceiling functions</a> are examples of integer-valued <a href="Function_of_a_real_variable" title="Function of a real variable">functions of a real variable</a>, but on <a href="Real_number" title="Real number">real numbers</a> and, generally, on (non-disconnected) <a href="Topological_space" title="Topological space">topological spaces</a> integer-valued functions are not especially useful. Any such function on a <a href="Connected_space" title="Connected space">connected space</a> either has <a href="Discontinuity_(mathematics)" class="mw-redirect" title="Discontinuity (mathematics)">discontinuities</a> or is <a href="Constant_function" title="Constant function">constant</a>. On the other hand, on <a href="Discrete_space" title="Discrete space">discrete</a> and other <a href="Totally_disconnected_space" title="Totally disconnected space">totally disconnected spaces</a> integer-valued functions have roughly the same importance as <a href="Real-valued_function" title="Real-valued function">real-valued functions</a> have on non-discrete spaces.
</p><p>Any function with <a href="Natural_number" title="Natural number">natural</a>, or <a href="Non-negative" class="mw-redirect" title="Non-negative">non-negative</a> integer values is a partial case of an integer-valued function.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>
<p>Integer-valued functions defined on the domain of all real numbers include the floor and ceiling functions, the <a href="Dirichlet_function" title="Dirichlet function">Dirichlet function</a>, the <a href="Sign_function" title="Sign function">sign function</a> and the <a href="Heaviside_step_function" title="Heaviside step function">Heaviside step function</a> (except possibly at 0).
</p><p>Integer-valued functions defined on the domain of non-negative real numbers include the <a href="Integer_square_root" title="Integer square root">integer square root</a> function and the <a href="Prime-counting_function" title="Prime-counting function">prime-counting function</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Algebraic_properties">Algebraic properties</h2></div>
<p>On an arbitrary <a href="Set_(mathematics)" title="Set (mathematics)">set</a> <span class="texhtml mvar" style="font-style:italic;">X</span>, integer-valued functions form a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> with <a href="Pointwise" title="Pointwise">pointwise</a> operations of addition and multiplication,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and also an <a href="Algebra_over_a_ring" class="mw-redirect" title="Algebra over a ring">algebra</a> over the ring <span class="texhtml"><b>Z</b></span> of integers. Since the latter is an <a href="Ordered_ring" title="Ordered ring">ordered ring</a>, the functions form a <a href="Partially_ordered_ring" title="Partially ordered ring">partially ordered ring</a>:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\leq g\quad \iff \quad \forall x:f(x)\leq g(x).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>≤<!-- ≤ --></mo>
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<mspace width="thickmathspace"></mspace>
<mo stretchy="false">⟺<!-- ⟺ --></mo>
<mspace width="thickmathspace"></mspace>
<mspace width="1em"></mspace>
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<mi>x</mi>
<mo>:</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
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<annotation encoding="application/x-tex">{\displaystyle f\leq g\quad \iff \quad \forall x:f(x)\leq g(x).}</annotation>
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</math></span><img src="./e7a5060f9aac12fcf713662c3da8f96e90b8a211.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.013ex; height:2.843ex;" alt="{\displaystyle f\leq g\quad \iff \quad \forall x:f(x)\leq g(x).}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Uses">Uses</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Graph_theory_and_algebra">Graph theory and algebra</h3></div>
<p>Integer-valued functions are ubiquitous in <a href="Graph_theory" title="Graph theory">graph theory</a>. They also have similar uses in <a href="Geometric_group_theory" title="Geometric group theory">geometric group theory</a>, where <i><a href="Length_function" title="Length function">length function</a></i> represents the concept of <a href="Norm_(mathematics)" title="Norm (mathematics)">norm</a>, and <i><a href="Word_metric" title="Word metric">word metric</a></i> represents the concept of <a href="Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a>.
</p><p><a href="Integer-valued_polynomial" title="Integer-valued polynomial">Integer-valued polynomials</a> are important in <a href="Ring_theory" title="Ring theory">ring theory</a>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Mathematical_logic_and_computability_theory">Mathematical logic and computability theory</h3></div>
<p>In <a href="Mathematical_logic" title="Mathematical logic">mathematical logic</a>, such concepts as <a href="Primitive_recursive_function" title="Primitive recursive function">primitive recursive functions</a> and <a href="%CE%9C-recursive_function" class="mw-redirect" title="Μ-recursive function">μ-recursive functions</a> represent integer-valued functions of several natural variables or, in other words, functions on <span class="texhtml"><b>N</b><sup><i>n</i></sup></span>. <a href="G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a>, defined on <a href="Well-formed_formula" title="Well-formed formula">well-formed formulae</a> of some <a href="Formal_language" title="Formal language">formal language</a>, is a natural-valued function.
</p><p><a href="Computability_theory" title="Computability theory">Computability theory</a> is essentially based on natural numbers and natural (or integer) functions on them.
</p>
<div class="mw-heading mw-heading3"><h3 id="Number_theory">Number theory</h3></div>
<p>In <a href="Number_theory" title="Number theory">number theory</a>, many <a href="Arithmetic_function" title="Arithmetic function">arithmetic functions</a> are integer-valued.
</p>
<div class="mw-heading mw-heading3"><h3 id="Computer_science">Computer science</h3></div>
<p>In <a href="Computer_programming" title="Computer programming">computer programming</a>, many <a href="Function_(programming)" class="mw-redirect" title="Function (programming)">functions</a> return values of <a href="Integer_(computer_science)" title="Integer (computer science)">integer type</a> due to simplicity of implementation.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Integer-valued_polynomial" title="Integer-valued polynomial">Integer-valued polynomial</a></li>
<li><a href="Semi-continuity" title="Semi-continuity">Semi-continuity</a></li>
<li><a href="Rank_(disambiguation)#Mathematics" class="mw-disambig" title="Rank (disambiguation)">Rank (disambiguation)#Mathematics</a></li>
<li><a href="Grade_(disambiguation)" class="mw-redirect mw-disambig" title="Grade (disambiguation)">Grade (disambiguation)#In mathematics</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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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="CITEREFDummitFoote2003" class="citation book cs1">Dummit, David S.; Foote, Richard M. (July 2003). <i>Abstract Algebra</i> (3rd ed.). John Wiley and Sons, Inc. p. 225. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-43334-7</bdi>.</cite></span>
</li>
</ol></div></div>
<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>
<ul><li><a rel="nofollow" class="external free" href="https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SurveyIntegerValuedEntireFunctions.pdf">https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SurveyIntegerValuedEntireFunctions.pdf</a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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