SherbyNet - /page/wikibook/entry.mu

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Smoothness

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</style><div role="note" class="hatnote navigation-not-searchable">"C infinity" redirects here. For the extended complex plane <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} _{\infty }}">

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

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</math><img src="./8f96a8d50c59283dd1856c1d3564fcec2cbfe45e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.553ex; height:2.509ex;" alt="{\displaystyle \mathbb {C} _{\infty }}" loading="lazy">, see <a href="Riemann_sphere" title="Riemann sphere">Riemann sphere</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">"C^n" redirects here. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}">

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

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</math><img src="./a53b4e76242764d1bca004168353c380fef25258.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}" loading="lazy">, see <a href="Complex_coordinate_space" title="Complex coordinate space">Complex coordinate space</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">For smoothness in number theory, see <a href="Smooth_number" title="Smooth number">smooth number</a>.</div>

In <a href="Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, the smoothness of a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> is a property measured by the number of <a href="Continuous_function" title="Continuous function">continuous</a> <a href="Derivative_(mathematics)" class="mw-redirect" title="Derivative (mathematics)">derivatives</a> (differentiability class) it has over its <a href="Domain_of_a_function" title="Domain of a function">domain</a>.<a href="#cite_note-1">[1]</a>

A function of class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}">

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

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</math><img src="./167fdb0cfb5644c4623b5842e1a9141acd83b534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}" loading="lazy"> is a function of smoothness at least k; that is, a function of class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}">

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

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</math><img src="./167fdb0cfb5644c4623b5842e1a9141acd83b534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}" loading="lazy"> is a function that has a kth derivative that is continuous in its domain.

A function of class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }}">

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

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</math><img src="./971ed05871d69309df32efdfd2020128c9cf69d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.343ex;" alt="{\displaystyle C^{\infty }}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }}">

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

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</math><img src="./971ed05871d69309df32efdfd2020128c9cf69d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.343ex;" alt="{\displaystyle C^{\infty }}" loading="lazy">-function (pronounced C-infinity function) is an infinitely differentiable function, that is, a function that has derivatives of all <a href="Order_of_derivation" class="mw-redirect" title="Order of derivation">orders</a> (this implies that all these derivatives are continuous).

Generally, the term smooth function refers to a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }}">

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

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</math><img src="./971ed05871d69309df32efdfd2020128c9cf69d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.343ex;" alt="{\displaystyle C^{\infty }}" loading="lazy">-function. However, it may also mean "sufficiently differentiable" for the problem under consideration.

<div class="mw-heading mw-heading2"><h2 id="Differentiability_classes">Differentiability classes</h2></div>

Differentiability class is a classification of functions according to the properties of their <a href="Derivative" title="Derivative">derivatives</a>. It is a measure of the highest order of derivative that exists and is continuous for a function.

Consider an <a href="Open_set" title="Open set">open set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> on the <a href="Real_line" class="mw-redirect" title="Real line">real line</a> and a function <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"> defined on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> with real values. Let k be a non-negative <a href="Integer" title="Integer">integer</a>. The function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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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"> is said to be of differentiability class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}">

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

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</math><img src="./167fdb0cfb5644c4623b5842e1a9141acd83b534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}" loading="lazy"> if the derivatives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f',f'',\dots ,f^{(k)}}">

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

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</math><img src="./a7fe894bf9e0a425a4330bd421514c75d0269289.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.364ex; height:3.176ex;" alt="{\displaystyle f',f'',\dots ,f^{(k)}}" loading="lazy"> exist and are <a href="Continuous_function" title="Continuous function">continuous</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U.}">

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

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</math><img src="./7a305ef479ab152035f334467a2c314baa23eb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.429ex; height:2.176ex;" alt="{\displaystyle U.}" loading="lazy"> If <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"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-differentiable on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U,}">

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

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</math><img src="./59c21c569e498e0197798e3428b2bc6f25c0a457.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.429ex; height:2.509ex;" alt="{\displaystyle U,}" loading="lazy"> then it is at least in the class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k-1}}">

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

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</math><img src="./598147cbe0cb4a76fd846d940f40f98054613692.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.987ex; height:2.676ex;" alt="{\displaystyle C^{k-1}}" loading="lazy"> since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f',f'',\dots ,f^{(k-1)}}">

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

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</math><img src="./ca1bdc1d87012c2143a5b5de3a335c86dea39c0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.464ex; height:3.176ex;" alt="{\displaystyle f',f'',\dots ,f^{(k-1)}}" loading="lazy"> are continuous on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U.}">

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

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</math><img src="./7a305ef479ab152035f334467a2c314baa23eb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.429ex; height:2.176ex;" alt="{\displaystyle U.}" loading="lazy"> The function <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"> is said to be infinitely differentiable, smooth, or of class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty },}">

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

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</math><img src="./16c7cf8322706166398a2c611d52b988ff0824b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.32ex; height:2.676ex;" alt="{\displaystyle C^{\infty },}" loading="lazy"> if it has derivatives of all orders on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U.}">

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

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</math><img src="./7a305ef479ab152035f334467a2c314baa23eb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.429ex; height:2.176ex;" alt="{\displaystyle U.}" loading="lazy"> (So all these derivatives are continuous functions over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U.}">

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

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</math><img src="./7a305ef479ab152035f334467a2c314baa23eb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.429ex; height:2.176ex;" alt="{\displaystyle U.}" loading="lazy">)<a href="#cite_note-def_diff-2">[2]</a> The function <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"> is said to be of class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\omega },}">

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

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</math><img src="./0f15087f3538946579b5d53935ce8752b0297e0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.699ex; height:2.676ex;" alt="{\displaystyle C^{\omega },}" loading="lazy"> or <a href="Analytic_function" title="Analytic function">analytic</a>, if <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"> is smooth (i.e., <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"> is in the class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }}">

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

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</math><img src="./971ed05871d69309df32efdfd2020128c9cf69d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.343ex;" alt="{\displaystyle C^{\infty }}" loading="lazy">) and its <a href="Taylor_series" title="Taylor series">Taylor series</a> expansion around any point in its domain converges to the function in some <a href="Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighborhood</a> of the point. There exist functions that are smooth but not analytic; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\omega }}">

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

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</math><img src="./38d37d5006f57a93d8c29b58421c2cb8f5473ad8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.052ex; height:2.343ex;" alt="{\displaystyle C^{\omega }}" loading="lazy"> is thus strictly contained in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }.}">

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

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</math><img src="./149f006dc6f05e8f662e839f68194d287119da68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.32ex; height:2.343ex;" alt="{\displaystyle C^{\infty }.}" loading="lazy"> <a href="Bump_function" title="Bump function">Bump functions</a> are examples of functions with this property.

To put it differently, the class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{0}}">

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

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</math><img src="./7c14274cad45f7c22b662e7a4e56b1db052883d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{0}}" loading="lazy"> consists of all continuous functions. The class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{1}}">

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

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</math><img src="./bd24bae0d7570018e828e19851902c09c618af91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{1}}" loading="lazy"> consists of all <a href="Differentiable_function" title="Differentiable function">differentiable functions</a> whose derivative is continuous; such functions are called continuously differentiable. Thus, a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{1}}">

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

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</math><img src="./bd24bae0d7570018e828e19851902c09c618af91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{1}}" loading="lazy"> function is exactly a function whose derivative exists and is of class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{0}.}">

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

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</math><img src="./a67cb0c98055d92fb0a74fa796de55cea6acb5b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.499ex; height:2.676ex;" alt="{\displaystyle C^{0}.}" loading="lazy"> In general, the classes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}">

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

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</math><img src="./167fdb0cfb5644c4623b5842e1a9141acd83b534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}" loading="lazy"> can be defined <a href="Recursion" title="Recursion">recursively</a> by declaring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{0}}">

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

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</math><img src="./7c14274cad45f7c22b662e7a4e56b1db052883d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{0}}" loading="lazy"> to be the set of all continuous functions, and declaring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}">

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

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</math><img src="./167fdb0cfb5644c4623b5842e1a9141acd83b534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}" loading="lazy"> for any positive integer <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="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> to be the set of all differentiable functions whose derivative is in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k-1}.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4bd3f19e0f11963da8fc3f0feea3f50cb34a8d37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.634ex; height:2.676ex;" alt="{\displaystyle C^{k-1}.}" loading="lazy"> In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./167fdb0cfb5644c4623b5842e1a9141acd83b534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}" loading="lazy"> is contained in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./598147cbe0cb4a76fd846d940f40f98054613692.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.987ex; height:2.676ex;" alt="{\displaystyle C^{k-1}}" loading="lazy"> for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./549d18cca65d346b3e92efdaac24eae22da4da35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.119ex; height:2.509ex;" alt="{\displaystyle k>0,}" loading="lazy"> and there are examples to show that this containment is strict (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}\subsetneq C^{k-1}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./50a7a44c345b4f6f101793c07c5414153f5f98de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.972ex; height:3.176ex;" alt="{\displaystyle C^{k}\subsetneq C^{k-1}}" loading="lazy">). The class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./971ed05871d69309df32efdfd2020128c9cf69d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.343ex;" alt="{\displaystyle C^{\infty }}" loading="lazy"> of infinitely differentiable functions, is the intersection of the classes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./167fdb0cfb5644c4623b5842e1a9141acd83b534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}" loading="lazy"> as <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="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> varies over the non-negative integers.

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

<div class="mw-heading mw-heading4"><h4 id="Example:_continuous_(C0)_but_not_differentiable">Example: continuous (C0) but not differentiable</h4></div>

The function

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mstyle>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math>

is continuous, but not differentiable at x = 0, so it is of class C0, but not of class C1.

<div class="mw-heading mw-heading4"><h4 id="Example:_finitely-times_differentiable_(Ck)">Example: finitely-times differentiable (Ck)</h4></div>

For each even integer k, the function

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math>

is continuous and k times differentiable at all x. At x = 0, however, <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 (k + 1) times differentiable, so <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 of class Ck, but not of class Cj where j > k.

<div class="mw-heading mw-heading4"><h4 id="Example:_differentiable_but_not_continuously_differentiable_(not_C1)">Example: differentiable but not continuously differentiable (not C1)</h4></div>

The function

<mrow>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)={\begin{cases}x^{2}\sin {\left({\tfrac {1}{x}}\right)}&{\text{if }}x\neq 0,\\0&{\text{if }}x=0\end{cases}}}</annotation>

</semantics>

</math>

is differentiable, with derivative

<msup>

</msup>

<mrow>

<mtr>

<mtd>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g'(x)={\begin{cases}-{\mathord {\cos \left({\tfrac {1}{x}}\right)}}+2x\sin \left({\tfrac {1}{x}}\right)&{\text{if }}x\neq 0,\\0&{\text{if }}x=0.\end{cases}}}</annotation>

</semantics>

</math>

Because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(1/x)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9db386f96e90bab936fcf74ca97643a588b9d243.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.575ex; height:2.843ex;" alt="{\displaystyle \cos(1/x)}" loading="lazy"> oscillates as x → 0, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g'(x)}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./25db4e8f365c512d07b7ae79f36bca83cd9c57d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.942ex; height:3.009ex;" alt="{\displaystyle g'(x)}" loading="lazy"> is not continuous at zero. Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" loading="lazy"> is differentiable but not of class C1.

<div class="mw-heading mw-heading4"><h4 id="Example:_differentiable_but_not_Lipschitz_continuous">Example: differentiable but not Lipschitz continuous</h4></div>

The function

<mrow>

<mtr>

<mtd>

<msup>

</mrow>

</mrow>

</msup>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</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}x^{4/3}\sin {\left({\tfrac {1}{x}}\right)}&{\text{if }}x\neq 0,\\0&{\text{if }}x=0\end{cases}}}</annotation>

</semantics>

</math>

is differentiable but its derivative is unbounded on a <a href="Compact_set" class="mw-redirect" title="Compact set">compact set</a>. Therefore, <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="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> is an example of a function that is differentiable but not locally <a href="Lipschitz_continuous" class="mw-redirect" title="Lipschitz continuous">Lipschitz continuous</a>.

<div class="mw-heading mw-heading4"><h4 id="Example:_analytic_(Cω)">Example: analytic (Cω)</h4></div>

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 <a href="Analytic_function" title="Analytic function">analytic</a>, and hence falls into the class Cω (where ω is the smallest <a href="Transfinite_ordinal" class="mw-redirect" title="Transfinite ordinal">transfinite ordinal</a>). The <a href="Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a> are also analytic wherever they are defined, because they are <a href="Trigonometric_functions#Euler's_formula_and_the_exponential_function" title="Trigonometric functions"> linear combinations of complex exponential functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{ix}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5271aa5bef13f6bd715afeda45bc59ae37d7c6d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.823ex; height:2.676ex;" alt="{\displaystyle e^{ix}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-ix}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./73d5a52f7846c68352bd1c9902f4ea65b68be970.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.102ex; height:2.676ex;" alt="{\displaystyle e^{-ix}}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Example:_smooth_(C∞)_but_not_analytic_(Cω)">Example: smooth (C∞) but not analytic (Cω)</h4></div>

The <a href="Bump_function" title="Bump function">bump function</a>

<mrow>

<mtr>

<mtd>

<msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</msup>

</mtd>

<mtd>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext> otherwise </mtext>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)={\begin{cases}e^{-{\frac {1}{1-x^{2}}}}&{\text{ if }}|x|<1,\\0&{\text{ otherwise }}\end{cases}}}</annotation>

</semantics>

</math>

is smooth, so of class C∞, but it is not analytic at x = ±1, and hence is not of class Cω. The function f is an example of a smooth function with <a href="Compact_support" class="mw-redirect" title="Compact support">compact support</a>.

<div class="mw-heading mw-heading3"><h3 id="Multivariate_differentiability_classes">Multivariate differentiability classes</h3></div>

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

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8c887467347dd916da3a7f883b0f302a71a8583a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.285ex; height:2.676ex;" alt="{\displaystyle f:U\subseteq \mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"> defined on an open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> is said<a href="#cite_note-3">[3]</a> to be of class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./167fdb0cfb5644c4623b5842e1a9141acd83b534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">, for a positive integer <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="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">, if all <a href="Partial_derivatives" class="mw-redirect" title="Partial derivatives">partial derivatives</a>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{\alpha }f}{\partial x_{1}^{\alpha _{1}}\,\partial x_{2}^{\alpha _{2}}\,\cdots \,\partial x_{n}^{\alpha _{n}}}}(y_{1},y_{2},\ldots ,y_{n})}</annotation>

</semantics>

</math>

exist and are continuous, for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}}</annotation>

</semantics>

</math><img src="./c64865c1adff5bac6df9d9751bebd80796115b3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.002ex; height:2.009ex;" alt="{\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}}" loading="lazy"> non-negative integers, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}\leq k}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha =\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}\leq k}</annotation>

</semantics>

</math><img src="./22eb260c7d7b3af3fa6648398c46cffbd0de5d61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.93ex; height:2.509ex;" alt="{\displaystyle \alpha =\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}\leq k}" loading="lazy">, and every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y_{1},y_{2},\ldots ,y_{n})\in U}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (y_{1},y_{2},\ldots ,y_{n})\in U}</annotation>

</semantics>

</math><img src="./6eff6c450266b8761e32e882461ff73f2a5b39a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.389ex; height:2.843ex;" alt="{\displaystyle (y_{1},y_{2},\ldots ,y_{n})\in U}" loading="lazy">. Equivalently, <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 of class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./167fdb0cfb5644c4623b5842e1a9141acd83b534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> if the <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="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-th order <a href="Fr%C3%A9chet_derivative" title="Fréchet derivative">Fréchet derivative</a> 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"> exists and is continuous at every point of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">. 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 said to be of class <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"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{0}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7c14274cad45f7c22b662e7a4e56b1db052883d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{0}}" loading="lazy"> if it is continuous on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">. Functions of class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./bd24bae0d7570018e828e19851902c09c618af91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{1}}" loading="lazy"> are also said to be continuously differentiable.

A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:U\subset \mathbb {R} ^{n}\to \mathbb {R} ^{m}}">

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:U\subset \mathbb {R} ^{n}\to \mathbb {R} ^{m}}</annotation>

</semantics>

</math><img src="./4a44f98a46e1ff516318e4203b61a15cf96ec16f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.961ex; height:2.676ex;" alt="{\displaystyle f:U\subset \mathbb {R} ^{n}\to \mathbb {R} ^{m}}" loading="lazy">, defined on an open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy">, is said to be of class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./167fdb0cfb5644c4623b5842e1a9141acd83b534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">, for a positive integer <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="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">, if all of its components

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{i}(x_{1},x_{2},\ldots ,x_{n})=(\pi _{i}\circ f)(x_{1},x_{2},\ldots ,x_{n})=\pi _{i}(f(x_{1},x_{2},\ldots ,x_{n})){\text{ for }}i=1,2,3,\ldots ,m}</annotation>

</semantics>

</math>

are of class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./167fdb0cfb5644c4623b5842e1a9141acd83b534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./51bb06f24c5ba78c53df3a04cb60ba05ef6f08aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.125ex; height:2.009ex;" alt="{\displaystyle \pi _{i}}" loading="lazy"> are the natural <a href="Projection_(linear_algebra)" title="Projection (linear algebra)">projections</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{i}:\mathbb {R} ^{m}\to \mathbb {R} }">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi _{i}:\mathbb {R} ^{m}\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./e3c99ce679c4f230f45fcc7ee59b4170cdf3597e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.707ex; height:2.676ex;" alt="{\displaystyle \pi _{i}:\mathbb {R} ^{m}\to \mathbb {R} }" loading="lazy"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{i}(x_{1},x_{2},\ldots ,x_{m})=x_{i}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi _{i}(x_{1},x_{2},\ldots ,x_{m})=x_{i}}</annotation>

</semantics>

</math><img src="./27c0c00a15a4bed743debf01dab1335f56c3e2f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.147ex; height:2.843ex;" alt="{\displaystyle \pi _{i}(x_{1},x_{2},\ldots ,x_{m})=x_{i}}" loading="lazy">. It is said to be of class <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"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{0}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7c14274cad45f7c22b662e7a4e56b1db052883d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{0}}" loading="lazy"> if it is continuous, or equivalently, if all components <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./65da883ca3d16b461e46c94777b0d9c4aa010e79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}" loading="lazy"> are continuous, on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="The_space_of_Ck_functions">The space of Ck functions</h3></div>

Let <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"> be an open subset of the real line. The set of all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./167fdb0cfb5644c4623b5842e1a9141acd83b534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}" loading="lazy"> real-valued functions defined 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"> is a <a href="Fr%C3%A9chet_space" title="Fréchet space">Fréchet vector space</a>, with the countable family of <a href="Seminorm" title="Seminorm">seminorms</a>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{K,m}=\sup _{x\in K}\left|f^{(m)}(x)\right|}</annotation>

</semantics>

</math>

where <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"> varies over an increasing sequence of <a href="Compact_set" class="mw-redirect" title="Compact set">compact sets</a> whose <a href="Union_(set_theory)" title="Union (set theory)">union</a> is <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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=0,1,\dots ,k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m=0,1,\dots ,k}</annotation>

</semantics>

</math><img src="./e610d667a3f31042e7c154d117ffdfe1af139a7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.887ex; height:2.509ex;" alt="{\displaystyle m=0,1,\dots ,k}" loading="lazy">.

The set of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./971ed05871d69309df32efdfd2020128c9cf69d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.343ex;" alt="{\displaystyle C^{\infty }}" loading="lazy"> functions over <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"> also forms a Fréchet space. One uses the same seminorms as above, except that <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="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> is allowed to range over all non-negative integer values.

The above spaces occur naturally in applications where functions having derivatives of certain orders are necessary; however, particularly in the study of <a href="Partial_differential_equation" title="Partial differential equation">partial differential equations</a>, it can sometimes be more fruitful to work instead with the <a href="Sobolev_space" title="Sobolev space">Sobolev spaces</a>.

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

The terms parametric continuity (Ck) and geometric continuity (Gn) were introduced by <a href="Brian_A._Barsky" title="Brian A. Barsky">Brian Barsky</a>, to show that the smoothness of a curve could be measured by removing restrictions on the <a href="Speed" title="Speed">speed</a>, with which the parameter traces out the curve.<a href="#cite_note-Barsky1981-4">[4]</a><a href="#cite_note-Barsky1988-5">[5]</a><a href="#cite_note-BartelsBeattyBarsky1987-6">[6]</a>

<div class="mw-heading mw-heading3"><h3 id="Parametric_continuity">Parametric continuity</h3></div>

Parametric continuity (Ck) is a concept applied to <a href="Parametric_curve" class="mw-redirect" title="Parametric curve">parametric curves</a>, which describes the smoothness of the parameter's value with distance along the curve. A (parametric) curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s:[0,1]\to \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s:[0,1]\to \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./876c4c194673fe6a8d7d9c0855101f30f1c5c4df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.191ex; height:2.843ex;" alt="{\displaystyle s:[0,1]\to \mathbb {R} ^{n}}" loading="lazy"> is said to be of class Ck, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {d^{k}s}{dt^{k}}}}">

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {d^{k}s}{dt^{k}}}}</annotation>

</semantics>

</math><img src="./cdf0e37a1c6213bf435eee76948b5a39a1ffe312.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:3.328ex; height:4.343ex;" alt="{\displaystyle \textstyle {\frac {d^{k}s}{dt^{k}}}}" loading="lazy"> exists and is continuous on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation>

</semantics>

</math><img src="./738f7d23bb2d9642bab520020873cccbef49768d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}" loading="lazy">, where derivatives at the end-points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

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

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</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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

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

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.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"> are taken to be <a href="Semi-differentiability" title="Semi-differentiability">one sided derivatives</a> (from the right at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

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

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</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"> and from the left at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

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</mrow>

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

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</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.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">).

As a practical application of this concept, a curve describing the motion of an object with a parameter of time must have C1 continuity and its first derivative is differentiable—for the object to have finite acceleration. For smoother motion, such as that of a camera's path while making a film, higher orders of parametric continuity are required.

<div class="mw-heading mw-heading4"><h4 id="Order_of_parametric_continuity">Order of parametric continuity</h4></div>

The various order of parametric continuity can be described as follows:<a href="#cite_note-7">[7]</a>

<msup>

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

</semantics>

</math><img src="./7c14274cad45f7c22b662e7a4e56b1db052883d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{0}}" loading="lazy">: zeroth derivative is continuous (curves are continuous)</li>

<msup>

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

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</math><img src="./bd24bae0d7570018e828e19851902c09c618af91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{1}}" loading="lazy">: zeroth and first derivatives are continuous</li>

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

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</math><img src="./4fd6a5946b7e916352b0afc557f992328bac85e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{2}}" loading="lazy">: zeroth, first and second derivatives are continuous</li>

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

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</math><img src="./50cc4309121472bbd901a7e54c365829cd150d82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.016ex; height:2.343ex;" alt="{\displaystyle C^{n}}" loading="lazy">: 0-th through <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

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

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</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 derivatives are continuous</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Geometric_continuity">Geometric continuity</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Geometrical_continuity" title="Geometrical continuity">Geometrical continuity</a>.</div>

A <a href="Curve" title="Curve">curve</a> or <a href="Surface_(topology)" title="Surface (topology)">surface</a> can be described as having <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{n}}">

<msup>

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

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</math><img src="./f43241538481dd8112f3c0ada4237a2c7b2aa6e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.045ex; height:2.343ex;" alt="{\displaystyle G^{n}}" loading="lazy"> continuity, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

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<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"> being the increasing measure of smoothness. Consider the segments either side of a point on a curve:

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<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">: The curves touch at the join point.</li>

<msup>

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

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</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">: The curves also share a common <a href="Tangent" title="Tangent">tangent</a> direction at the join point.</li>

<msup>

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<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">: The curves also share a common center of curvature at the join point.</li></ul>

In general, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

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</math><img src="./f43241538481dd8112f3c0ada4237a2c7b2aa6e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.045ex; height:2.343ex;" alt="{\displaystyle G^{n}}" loading="lazy"> continuity exists if the curves can be reparameterized to have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{n}}">

<msup>

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

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</math><img src="./50cc4309121472bbd901a7e54c365829cd150d82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.016ex; height:2.343ex;" alt="{\displaystyle C^{n}}" loading="lazy"> (parametric) continuity.<a href="#cite_note-Barsky-DeRose-8">[8]</a><a href="#cite_note-9">[9]</a> A reparametrization of the curve is geometrically identical to the original; only the parameter is affected.

Equivalently, two vector functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}">

</mstyle>

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

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</math><img src="./5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t)}">

</mstyle>

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

</semantics>

</math><img src="./b84f700860ee7af27797d11ddfad3d185eb7af0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.765ex; height:2.843ex;" alt="{\displaystyle g(t)}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(1)=g(0)}">

</mstyle>

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

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</math><img src="./2a9ad3022c1f42dfd1bbfacb00c7162106e77e44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.437ex; height:2.843ex;" alt="{\displaystyle f(1)=g(0)}" loading="lazy"> have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

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</math><img src="./f43241538481dd8112f3c0ada4237a2c7b2aa6e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.045ex; height:2.343ex;" alt="{\displaystyle G^{n}}" loading="lazy"> continuity at the point where they meet if

they satisfy equations known as Beta-constraints. For example, the Beta-constraints for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{4}}">

<msup>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle G^{4}}</annotation>

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</math><img src="./e255387d6a1c952c0890cae141e625a4f597c895.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^{4}}" loading="lazy"> continuity are:

<mtr>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g^{(1)}(0)&=\beta _{1}f^{(1)}(1)\\g^{(2)}(0)&=\beta _{1}^{2}f^{(2)}(1)+\beta _{2}f^{(1)}(1)\\g^{(3)}(0)&=\beta _{1}^{3}f^{(3)}(1)+3\beta _{1}\beta _{2}f^{(2)}(1)+\beta _{3}f^{(1)}(1)\\g^{(4)}(0)&=\beta _{1}^{4}f^{(4)}(1)+6\beta _{1}^{2}\beta _{2}f^{(3)}(1)+(4\beta _{1}\beta _{3}+3\beta _{2}^{2})f^{(2)}(1)+\beta _{4}f^{(1)}(1)\\\end{aligned}}}</annotation>

</semantics>

</math><img src="./578f251aecaeb18a639dcf58345974e1b0f668a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; margin-top: -0.297ex; width:70.089ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}g^{(1)}(0)&=\beta _{1}f^{(1)}(1)\\g^{(2)}(0)&=\beta _{1}^{2}f^{(2)}(1)+\beta _{2}f^{(1)}(1)\\g^{(3)}(0)&=\beta _{1}^{3}f^{(3)}(1)+3\beta _{1}\beta _{2}f^{(2)}(1)+\beta _{3}f^{(1)}(1)\\g^{(4)}(0)&=\beta _{1}^{4}f^{(4)}(1)+6\beta _{1}^{2}\beta _{2}f^{(3)}(1)+(4\beta _{1}\beta _{3}+3\beta _{2}^{2})f^{(2)}(1)+\beta _{4}f^{(1)}(1)\\\end{aligned}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{2}}">

<msub>

</mrow>

</msub>

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

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</math><img src="./8d30285b40d7488ae6caef3beb7106142869fbea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{2}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{3}}</annotation>

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</math><img src="./5b1001bcd6968314a965726f5dd193d1b11ada59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{3}}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{4}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{4}}</annotation>

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</math><img src="./16fb7d2f7d9b9f310820b4e110b084003aa80fb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{4}}" loading="lazy"> are arbitrary, but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

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</math><img src="./eeeccd8b585b819e38f9c1fe5e9816a3ea01804c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{1}}" loading="lazy"> is constrained to be positive.<a href="#cite_note-Barsky-DeRose-8">[8]</a>: 65

In the case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}" loading="lazy">, this reduces to

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c60a84b61f8c7d3af304ea05667a2e1f40ab6ffc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.238ex; height:3.009ex;" alt="{\displaystyle f'(1)\neq 0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(1)=kg'(0)}">

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a54fbf3b88e8c40fd8dc83060e221dbd6c67cd4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.061ex; height:3.009ex;" alt="{\displaystyle f'(1)=kg'(0)}" loading="lazy">, for a scalar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./27b3af208b148139eefc03f0f80fa94c38c5af45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>0}" loading="lazy"> (i.e., the direction, but not necessarily the magnitude, of the two vectors is equal).

While it may be obvious that a curve would require <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 to appear smooth, for good <a href="Aesthetics" title="Aesthetics">aesthetics</a>, such as those aspired to in <a href="Architecture" title="Architecture">architecture</a> and <a href="Sports_car" title="Sports car">sports car</a> design, higher levels of geometric continuity are required. For example, reflections in a car body will not appear smooth unless the body has <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>

</semantics>

</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.

A <a href="Rounded_rectangle" class="mw-redirect" title="Rounded rectangle">rounded rectangle</a> (with ninety degree circular arcs at the four corners) has <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, but does not have <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>

</semantics>

</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. The same is true for a <a href="Rounded_cube" class="mw-redirect" title="Rounded cube">rounded cube</a>, with octants of a sphere at its corners and quarter-cylinders along its edges. If an editable curve with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{2}}">

<msup>

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

</semantics>

</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 is required, then <a href="Cubic_splines" class="mw-redirect" title="Cubic splines">cubic splines</a> are typically chosen; these curves are frequently used in <a href="Industrial_design" title="Industrial design">industrial design</a>.

<div class="mw-heading mw-heading2"><h2 id="Other_concepts">Other concepts</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Relation_to_analyticity">Relation to analyticity</h3></div>

While all <a href="Analytic_function" title="Analytic function">analytic functions</a> are "smooth" (i.e. have all derivatives continuous) on the set on which they are analytic, examples such as <a href="Bump_function" title="Bump function">bump functions</a> (mentioned above) show that the converse is not true for functions on the reals: there exist smooth real functions that are not analytic. Simple examples of functions that are <a href="Non-analytic_smooth_function#A_smooth_function_which_is_nowhere_real_analytic" title="Non-analytic smooth function">smooth but not analytic at any point</a> can be made by means of <a href="Fourier_series" title="Fourier series">Fourier series</a>; another example is the <a href="Fabius_function" title="Fabius function">Fabius function</a>. Although it might seem that such functions are the exception rather than the rule, it turns out that the analytic functions are scattered very thinly among the smooth ones; more rigorously, the analytic functions form a <a href="Meagre_set" title="Meagre set">meagre</a> subset of the smooth functions. Furthermore, for every open subset A of the real line, there exist smooth functions that are analytic on A and nowhere else.

It is useful to compare the situation to that of the ubiquity of <a href="Transcendental_number" title="Transcendental number">transcendental numbers</a> on the real line. Both on the real line and the set of smooth functions, the examples we come up with at first thought (algebraic/rational numbers and analytic functions) are far better behaved than the majority of cases: the transcendental numbers and nowhere analytic functions have full measure (their complements are meagre).

The situation thus described is in marked contrast to complex differentiable functions. If a complex function is differentiable just once on an open set, it is both infinitely differentiable and analytic on that set.

<div class="mw-heading mw-heading3"><h3 id="Smooth_partitions_of_unity">Smooth partitions of unity</h3></div>

Smooth functions with given closed <a href="Support_(mathematics)" title="Support (mathematics)">support</a> are used in the construction of smooth partitions of unity (see <a href="Partition_of_unity" title="Partition of unity">partition of unity</a> and <a href="Topology_glossary" class="mw-redirect" title="Topology glossary">topology glossary</a>); these are essential in the study of <a href="Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifolds</a>, for example to show that <a href="Riemannian_metric" class="mw-redirect" title="Riemannian metric">Riemannian metrics</a> can be defined globally starting from their local existence. A simple case is that of a <a href="Bump_function" title="Bump function">bump function</a> on the real line, that is, a smooth function f that takes the value 0 outside an interval [a,b] and such that

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)>0\quad {\text{ for }}\quad a<x<b.\,}</annotation>

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</math>

Given a number of overlapping intervals on the line, bump functions can be constructed on each of them, and on semi-infinite intervals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,c]}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./41438072d1ed991929e008e55b7e45d1a721af42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.724ex; height:2.843ex;" alt="{\displaystyle (-\infty ,c]}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [d,+\infty )}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1bf7d5f58387e41a9aba35f0389baf31156aa8f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.933ex; height:2.843ex;" alt="{\displaystyle [d,+\infty )}" loading="lazy"> to cover the whole line, such that the sum of the functions is always 1.

From what has just been said, partitions of unity do not apply to <a href="Holomorphic_function" title="Holomorphic function">holomorphic functions</a>; their different behavior relative to existence and <a href="Analytic_continuation" title="Analytic continuation">analytic continuation</a> is one of the roots of <a href="Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaf</a> theory. In contrast, sheaves of smooth functions tend not to carry much topological information.

<div class="mw-heading mw-heading3"><h3 id="Smooth_functions_on_and_between_manifolds">Smooth functions on and between manifolds</h3></div>

Given a <a href="Differentiable_manifold" title="Differentiable manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</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">, of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,}">

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</mrow>

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

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</math><img src="./dad66d19bb37bc69223cb004be2ea5dd95f9564c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.687ex; height:2.009ex;" alt="{\displaystyle m,}" loading="lazy"> and an <a href="Atlas_(topology)" title="Atlas (topology)">atlas</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {U}}=\{(U_{\alpha },\phi _{\alpha })\}_{\alpha },}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {U}}=\{(U_{\alpha },\phi _{\alpha })\}_{\alpha },}</annotation>

</semantics>

</math><img src="./7f2c1f8399bff60f7d7b1c6a3ebaf9299b00730f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.057ex; width:17.341ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {U}}=\{(U_{\alpha },\phi _{\alpha })\}_{\alpha },}" loading="lazy"> then a map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:M\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3f7c129dfb3446a2251aabc1cb3eba0b13ee7b21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.95ex; height:2.509ex;" alt="{\displaystyle f:M\to \mathbb {R} }" loading="lazy"> is smooth on <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"> if for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./35ad2c18a15749505c928763cd4fdb56f4982816.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.542ex; height:2.509ex;" alt="{\displaystyle p\in M}" loading="lazy"> there exists a chart <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U,\phi )\in {\mathfrak {U}},}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (U,\phi )\in {\mathfrak {U}},}</annotation>

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</math><img src="./0d45344f059d2bb712457369649e0565c61528eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.044ex; height:2.843ex;" alt="{\displaystyle (U,\phi )\in {\mathfrak {U}},}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in U,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2278b928cabc0c565326b6be2a725b44ae884347.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.529ex; height:2.509ex;" alt="{\displaystyle p\in U,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ \phi ^{-1}:\phi (U)\to \mathbb {R} }">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\circ \phi ^{-1}:\phi (U)\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./7d75a24c222b8c3eaa8a782f4795ac5c14da84b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.398ex; height:3.176ex;" alt="{\displaystyle f\circ \phi ^{-1}:\phi (U)\to \mathbb {R} }" loading="lazy"> is a smooth function from a neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (p)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./772d7fe1c5a3439845a87f55d45af038956b3f12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.364ex; height:2.843ex;" alt="{\displaystyle \phi (p)}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{m}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

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</math><img src="./6a87a024931038d1858dc22e8a194e5978c3412e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.353ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{m}}" loading="lazy"> to <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"> (all partial derivatives up to a given order are continuous). Smoothness can be checked with respect to any <a href="Chart_(topology)" class="mw-redirect" title="Chart (topology)">chart</a> of the atlas that contains <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./393fcf18074cb42eafb26b76c515a1e93e17512c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.906ex; height:2.009ex;" alt="{\displaystyle p,}" loading="lazy"> since the smoothness requirements on the transition functions between charts ensure that if <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 smooth near <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> in one chart it will be smooth near <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> in any other chart.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:M\to N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F:M\to N}</annotation>

</semantics>

</math><img src="./6d67a9ff9bb56216707b351d2d51afc2ff5fb034.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.798ex; height:2.176ex;" alt="{\displaystyle F:M\to N}" loading="lazy"> is a map from <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"> to an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</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">-dimensional manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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</mrow>

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

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.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 N}" loading="lazy">, 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="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> is smooth if, for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in M,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./691155b373b307bb2573745e5558b2b26596cf48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.189ex; height:2.509ex;" alt="{\displaystyle p\in M,}" loading="lazy"> there is a chart <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U,\phi )}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0be3491c45cdc56c113de78c13f5cb0b70d4c6d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.011ex; height:2.843ex;" alt="{\displaystyle (U,\phi )}" loading="lazy"> containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./393fcf18074cb42eafb26b76c515a1e93e17512c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.906ex; height:2.009ex;" alt="{\displaystyle p,}" loading="lazy"> and a chart <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (V,\psi )}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a954730c89ccab8dd8fa953dea39874780c52070.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.144ex; height:2.843ex;" alt="{\displaystyle (V,\psi )}" loading="lazy"> containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(p)}">

</mstyle>

</mrow>

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

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</math><img src="./f96aa8fc46e2bc3f1a5a1f612c3af35609d5802d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.72ex; height:2.843ex;" alt="{\displaystyle F(p)}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(U)\subset V,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7dc068fca966450e039cf655141fdfd8d3d39108.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.865ex; height:2.843ex;" alt="{\displaystyle F(U)\subset V,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \circ F\circ \phi ^{-1}:\phi (U)\to \psi (V)}">

<msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \psi \circ F\circ \phi ^{-1}:\phi (U)\to \psi (V)}</annotation>

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</math><img src="./2cbef11c4b8446487ce7384f2027396304e83f88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27ex; height:3.176ex;" alt="{\displaystyle \psi \circ F\circ \phi ^{-1}:\phi (U)\to \psi (V)}" loading="lazy"> is a smooth function from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{m}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6a87a024931038d1858dc22e8a194e5978c3412e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.353ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{m}}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}.}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./76ef548febfc9981762740107858be9e3a5576c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.543ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}.}" loading="lazy">

Smooth maps between manifolds induce linear maps between <a href="Tangent_space" title="Tangent space">tangent spaces</a>: for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:M\to N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F:M\to N}</annotation>

</semantics>

</math><img src="./6d67a9ff9bb56216707b351d2d51afc2ff5fb034.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.798ex; height:2.176ex;" alt="{\displaystyle F:M\to N}" loading="lazy">, at each point the <a href="Pushforward_(differential)" title="Pushforward (differential)">pushforward</a> (or differential) maps tangent vectors at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> to tangent vectors at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(p)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f96aa8fc46e2bc3f1a5a1f612c3af35609d5802d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.72ex; height:2.843ex;" alt="{\displaystyle F(p)}" loading="lazy">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{*,p}:T_{p}M\to T_{F(p)}N,}">

<msub>

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</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle F_{*,p}:T_{p}M\to T_{F(p)}N,}</annotation>

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</math><img src="./3277f7bd171faed258f776346f699516393c024c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.881ex; height:3.009ex;" alt="{\displaystyle F_{*,p}:T_{p}M\to T_{F(p)}N,}" loading="lazy"> and on the level of the <a href="Tangent_bundle" title="Tangent bundle">tangent bundle</a>, the pushforward is a <a href="Vector_bundle_homomorphism" class="mw-redirect" title="Vector bundle homomorphism">vector bundle homomorphism</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{*}:TM\to TN.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{*}:TM\to TN.}</annotation>

</semantics>

</math><img src="./b3bd2cfdd2b16c2b28ece06ba183526fe7296a02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.293ex; margin-bottom: -0.379ex; width:16.525ex; height:2.509ex;" alt="{\displaystyle F_{*}:TM\to TN.}" loading="lazy"> The dual to the pushforward is the <a href="Pullback_(differential_geometry)" title="Pullback (differential geometry)">pullback</a>, which "pulls" covectors on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.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 N}" loading="lazy"> back to covectors on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,}">

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

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</math><img src="./b466e90209f39c0c2caad1b11445824b82c2f536.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.089ex; height:2.509ex;" alt="{\displaystyle M,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-forms to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-forms: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{*}:\Omega ^{k}(N)\to \Omega ^{k}(M).}">

<msup>

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</msup>

<msup>

</mrow>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle F^{*}:\Omega ^{k}(N)\to \Omega ^{k}(M).}</annotation>

</semantics>

</math><img src="./717b57538fb8529b8b6ca9d2dcc6ecffaa62d2f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.725ex; height:3.176ex;" alt="{\displaystyle F^{*}:\Omega ^{k}(N)\to \Omega ^{k}(M).}" loading="lazy"> In this way smooth functions between manifolds can transport <a href="Sheaf_(mathematics)" title="Sheaf (mathematics)">local data</a>, like <a href="Vector_field" title="Vector field">vector fields</a> and <a href="Differential_form" title="Differential form">differential forms</a>, from one manifold to another, or down to Euclidean space where computations like <a href="Integration_on_manifolds" class="mw-redirect" title="Integration on manifolds">integration</a> are well understood.

Preimages and pushforwards along smooth functions are, in general, not manifolds without additional assumptions. Preimages of regular points (that is, if the differential does not vanish on the preimage) are manifolds; this is the <a href="Preimage_theorem" title="Preimage theorem">preimage theorem</a>. Similarly, pushforwards along embeddings are manifolds.<a href="#cite_note-10">[10]</a>

<div class="mw-heading mw-heading3"><h3 id="Smooth_functions_between_subsets_of_manifolds">Smooth functions between subsets of manifolds</h3></div>

There is a corresponding notion of smooth map for arbitrary subsets of manifolds. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

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

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</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 <a href="Function_(mathematics)" title="Function (mathematics)">function</a> whose <a href="Domain_of_a_function" title="Domain of a function">domain</a> and <a href="Range_of_a_function" title="Range of a function">range</a> are subsets of manifolds <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\subseteq M}">

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

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</math><img src="./4a8f625b2e9c810c24a95f54c8903345ba545235.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.521ex; height:2.343ex;" alt="{\displaystyle X\subseteq M}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\subseteq N}">

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

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</math><img src="./d53bc1ba03712f3a867809965ba50f7c831a958d.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 Y\subseteq N}" loading="lazy"> respectively. <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"> is said to be smooth if for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

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

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</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"> there is an open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subseteq M}">

</mstyle>

</mrow>

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

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</math><img src="./a57ecb017c8458e5f519dbca5ddb7fffde7f1a16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.323ex; height:2.343ex;" alt="{\displaystyle U\subseteq M}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in U}">

</mstyle>

</mrow>

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

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</math><img src="./c32ddcb2941216f2980b950ce969dc15cba26906.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.953ex; height:2.176ex;" alt="{\displaystyle x\in U}" loading="lazy"> and a smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:U\to N}">

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<annotation encoding="application/x-tex">{\displaystyle F:U\to N}</annotation>

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</math><img src="./da284858c0fca68bc0ab9c4fef8ef51d802d831d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.138ex; height:2.176ex;" alt="{\displaystyle F:U\to N}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(p)=f(p)}">

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

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</math><img src="./7024501dc8b167a805769e7e301cfb0ee5a15ee1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.075ex; height:2.843ex;" alt="{\displaystyle F(p)=f(p)}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in U\cap X.}">

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</mrow>

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

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</math><img src="./9e96b975f76f7861e48409c044d11bebb3efd650.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.092ex; height:2.509ex;" alt="{\displaystyle p\in U\cap X.}" loading="lazy">

<ul><li><a href="Discontinuity_(mathematics)" class="mw-redirect" title="Discontinuity (mathematics)">Discontinuity</a> – Mathematical analysis of discontinuous pointsPages displaying short descriptions of redirect targets</li>

<li><a href="Hadamard's_lemma" title="Hadamard's lemma">Hadamard's lemma</a></li>

<li><a href="Non-analytic_smooth_function" title="Non-analytic smooth function">Non-analytic smooth function</a> – Mathematical functions which are smooth but not analytic</li>

<li><a href="Quasi-analytic_function" title="Quasi-analytic function">Quasi-analytic function</a></li>

<li><a href="Singularity_(mathematics)" title="Singularity (mathematics)">Singularity (mathematics)</a> – Point where a function, a curve or another mathematical object does not behave regularly</li>

<li><a href="Sinuosity" title="Sinuosity">Sinuosity</a> – Ratio of arc length and straight-line distance between two points on a wave-like function</li>

<li><a href="Smooth_point" class="mw-redirect" title="Smooth point">Smooth scheme</a></li>

<li><a href="Smooth_number" title="Smooth number">Smooth number</a> – Integer having only small prime factors (number theory)</li>

<li><a href="Smoothing" title="Smoothing">Smoothing</a> – Fitting an approximating function to data</li>

<li><a href="Spline_(mathematics)" title="Spline (mathematics)">Spline</a> – Mathematical function defined piecewise by polynomials</li>

<li><a href="Sobolev_mapping" title="Sobolev mapping">Sobolev mapping</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></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><div id="Manifolds_(Glossary,_List,_Category)274" style="font-size:114%;margin:0 4em"><a href="Manifold" title="Manifold">Manifolds</a> (<a href="Glossary_of_differential_geometry_and_topology" title="Glossary of differential geometry and topology">Glossary</a>, <a href="List_of_manifolds" title="List of manifolds">List</a>, Category)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</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="Topological_manifold" title="Topological manifold">Topological manifold</a>

<ul><li><a href="Atlas_(topology)" title="Atlas (topology)">Atlas</a></li></ul></li>

<li><a href="Differentiable_manifold" title="Differentiable manifold">Differentiable/Smooth manifold</a>

<ul><li><a href="Differential_structure" title="Differential structure">Differential structure</a></li>

<li><a href="Smooth_structure" title="Smooth structure">Smooth atlas</a></li></ul></li>

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

<li><a href="Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li>

<li><a href="Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li>

<li><a href="Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li>

<li><a href="Tangent_space" title="Tangent space">Tangent space</a></li>

<li><a href="Differential_form" title="Differential form">Differential form</a></li>

<li><a href="Vector_field" title="Vector field">Vector field</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results (list)</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="Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index</a></li>

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

<li><a href="Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">Frobenius</a></li>

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

<li><a href="Hopf%E2%80%93Rinow_theorem" title="Hopf–Rinow theorem">Hopf–Rinow</a></li>

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

<li><a href="Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"></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="Differentiable_curve" title="Differentiable curve">Curve</a></li>

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

<ul><li><a href="Local_diffeomorphism" title="Local diffeomorphism">Local</a></li></ul></li>

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

<li><a href="Exponential_map_(Riemannian_geometry)" title="Exponential map (Riemannian geometry)">Exponential map</a>

<ul><li><a href="Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">in Lie theory</a></li></ul></li>

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

<li><a href="Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li>

<li><a href="Integral_curve" title="Integral curve">Integral curve</a></li>

<li><a href="Lie_derivative" title="Lie derivative">Lie derivative</a></li>

<li><a href="Section_(fiber_bundle)" title="Section (fiber bundle)">Section</a></li>

<li><a href="Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of manifolds</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="Closed_manifold" title="Closed manifold">Closed</a></li>

<li><a href="Collapsing_manifold" title="Collapsing manifold">Collapsing</a></li>

<li><a href="Complete_manifold" title="Complete manifold">Complete</a></li>

<li>(<a href="Almost_complex_manifold" title="Almost complex manifold">Almost</a>) <a href="Complex_manifold" title="Complex manifold">Complex</a></li>

<li>(<a href="Almost-contact_manifold" title="Almost-contact manifold">Almost</a>) <a href="Contact_manifold" class="mw-redirect" title="Contact manifold">Contact</a></li>

<li><a href="Fibered_manifold" title="Fibered manifold">Fibered</a></li>

<li><a href="Finsler_manifold" title="Finsler manifold">Finsler</a></li>

<li>(<a href="Almost_flat_manifold" title="Almost flat manifold">Almost</a>) <a href="Flat_manifold" title="Flat manifold">Flat</a></li>

<li><a href="G-structure_on_a_manifold" title="G-structure on a manifold">G-structure</a></li>

<li><a href="Hadamard_manifold" title="Hadamard manifold">Hadamard</a></li>

<li><a href="Hermitian_manifold" title="Hermitian manifold">Hermitian</a></li>

<li><a href="Hyperbolic_manifold" title="Hyperbolic manifold">Hyperbolic</a></li>

<li><a href="K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a></li>

<li><a href="Kenmotsu_manifold" title="Kenmotsu manifold">Kenmotsu</a></li>

<li><a href="Lie_group" title="Lie group">Lie group</a>

<ul><li><a href="Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie algebra</a></li></ul></li>

<li><a href="Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">Manifold with boundary</a></li>

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

<li><a href="Orientability" title="Orientability">Oriented</a></li>

<li><a href="Parallelizable_manifold" title="Parallelizable manifold">Parallelizable</a></li>

<li><a href="Poisson_manifold" title="Poisson manifold">Poisson</a></li>

<li><a href="Prime_manifold" title="Prime manifold">Prime</a></li>

<li><a href="Quaternionic_manifold" title="Quaternionic manifold">Quaternionic</a></li>

<li><a href="Hypercomplex_manifold" title="Hypercomplex manifold">Hypercomplex</a></li>

<li>(<a href="Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo-</a>, <a href="Sub-Riemannian_manifold" title="Sub-Riemannian manifold">Sub-</a>) <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li>

<li><a href="Rizza_manifold" title="Rizza manifold">Rizza</a></li>

<li><a href="Stein_manifold" title="Stein manifold">Stein</a></li>

<li>(<a href="Almost_symplectic_manifold" title="Almost symplectic manifold">Almost</a>) <a href="Symplectic_manifold" title="Symplectic manifold">Symplectic</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Tensor" title="Tensor">Tensors</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Vectors</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="Distribution_(differential_geometry)" title="Distribution (differential geometry)">Distribution</a></li>

<li><a href="Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a></li>

<li><a href="Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li>

<li><a href="Tangent_space" title="Tangent space">Tangent space</a>

<ul><li><a href="Tangent_bundle" title="Tangent bundle">bundle</a></li></ul></li>

<li><a href="Torsion_tensor" title="Torsion tensor">Torsion</a></li>

<li><a href="Vector_field" title="Vector field">Vector field</a></li>

<li><a href="Vector_flow" title="Vector flow">Vector flow</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Covectors</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="Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed/Exact</a></li>

<li><a href="Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li>

<li><a href="Cotangent_space" title="Cotangent space">Cotangent space</a>

<ul><li><a href="Cotangent_bundle" title="Cotangent bundle">bundle</a></li></ul></li>

<li><a href="De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li>

<li><a href="Differential_form" title="Differential form">Differential form</a>

<ul><li><a href="Vector-valued_differential_form" title="Vector-valued differential form">Vector-valued</a></li></ul></li>

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

<li><a href="Interior_product" title="Interior product">Interior product</a></li>

<li><a href="Pullback_(differential_geometry)" title="Pullback (differential geometry)">Pullback</a></li>

<li><a href="Ricci_curvature" title="Ricci curvature">Ricci curvature</a>

<li><a href="Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li>

<li><a href="Tensor_field" title="Tensor field">Tensor field</a>

<ul><li><a href="Tensor_density" title="Tensor density">density</a></li></ul></li>

<li><a href="Volume_form" title="Volume form">Volume form</a></li>

<li><a href="Wedge_product" class="mw-redirect" title="Wedge product">Wedge product</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Fiber_bundle" title="Fiber bundle">Bundles</a></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="Adjoint_bundle" title="Adjoint bundle">Adjoint</a></li>

<li><a href="Affine_bundle" title="Affine bundle">Affine</a></li>

<li><a href="Associated_bundle" title="Associated bundle">Associated</a></li>

<li><a href="Cotangent_bundle" title="Cotangent bundle">Cotangent</a></li>

<li><a href="Fiber_bundle" title="Fiber bundle">Fiber</a></li>

<li>(<a href="Cofibration" title="Cofibration">Co-</a>) <a href="Fibration" title="Fibration">Fibration</a></li>

<li><a href="Lie_algebra_bundle" title="Lie algebra bundle">Lie algebra</a></li>

<li>(<a href="Stable_normal_bundle" title="Stable normal bundle">Stable</a>) <a href="Normal_bundle" title="Normal bundle">Normal</a></li>

<li><a href="Principal_bundle" title="Principal bundle">Principal</a></li>

<li><a href="Spinor_bundle" title="Spinor bundle">Spinor</a></li>

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

<li><a href="Tangent_bundle" title="Tangent bundle">Tangent</a></li>

<li><a href="Tensor_bundle" title="Tensor bundle">Tensor</a></li>

<li><a href="Vector_bundle" title="Vector bundle">Vector</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Connection_(mathematics)" title="Connection (mathematics)">Connections</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="Affine_connection" title="Affine connection">Affine</a></li>

<li><a href="Cartan_connection" title="Cartan connection">Cartan</a></li>

<li><a href="Ehresmann_connection" title="Ehresmann connection">Ehresmann</a></li>

<li><a href="Connection_(fibred_manifold)" title="Connection (fibred manifold)">Generalized</a></li>

<li><a href="Koszul_connection" class="mw-redirect" title="Koszul connection">Koszul</a></li>

<li><a href="Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita</a></li>

<li><a href="Connection_(principal_bundle)" title="Connection (principal bundle)">Principal</a></li>

<li><a href="Connection_(vector_bundle)" title="Connection (vector bundle)">Vector</a></li>

<li><a href="Parallel_transport" title="Parallel transport">Parallel transport</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</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="Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a></li>

<li><a href="Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li>

<li><a href="History_of_manifolds_and_varieties" title="History of manifolds and varieties">History</a></li>

<li><a href="Morse_theory" title="Morse theory">Morse theory</a></li>

<li><a href="Moving_frame" title="Moving frame">Moving frame</a></li>

<li><a href="Singularity_theory" title="Singularity theory">Singularity theory</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="Banach_manifold" title="Banach manifold">Banach manifold</a></li>

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

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

<li><a href="Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li>

<li><a href="K-theory" title="K-theory">K-theory</a></li>

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

<li><a href="Secondary_calculus_and_cohomological_physics" title="Secondary calculus and cohomological physics">Secondary calculus</a>

<ul><li><a href="Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">over commutative algebras</a></li></ul></li>

<li><a href="Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf</a></li>

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

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

<li><a href="Stratified_space" title="Stratified space">Stratified space</a></li></ul>

</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="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 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></div><div><div class="zim-footer">

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