SherbyNet - /page/wikibook/entry.mu

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Bump function

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In <a href="Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, a bump function (also called a test function) is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }">

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

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</math><img src="./306c097f43c91dce633d12cde024948d39e73752.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.404ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"> on a <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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

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</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"> which is both <a href="Smooth_function" class="mw-redirect" title="Smooth function">smooth</a> (in the sense of having <a href="Continuous_function" title="Continuous function">continuous</a> <a href="Derivative" title="Derivative">derivatives</a> of all orders) and <a href="Support_(mathematics)#Compact_support" title="Support (mathematics)">compactly supported</a>. The <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of all bump functions with <a href="Domain_of_a_function" title="Domain of a function">domain</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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

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</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"> forms a <a href="Vector_space" title="Vector space">vector space</a>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {C} _{0}^{\infty }(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {C} _{0}^{\infty }(\mathbb {R} ^{n})}</annotation>

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</math><img src="./5c42ad5e1bd5583c3f724800d5175f1ad4d8e0f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.259ex; height:2.843ex;" alt="{\displaystyle \mathrm {C} _{0}^{\infty }(\mathbb {R} ^{n})}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {C} _{\mathrm {c} }^{\infty }(\mathbb {R} ^{n}).}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {C} _{\mathrm {c} }^{\infty }(\mathbb {R} ^{n}).}</annotation>

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</math><img src="./4e7f95aa101a636985c636ec114ddd6920598fe7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.906ex; height:2.843ex;" alt="{\displaystyle \mathrm {C} _{\mathrm {c} }^{\infty }(\mathbb {R} ^{n}).}" loading="lazy"> The <a href="Dual_space" title="Dual space">dual space</a> of this space endowed with a suitable <a href="Topological_space#Definitions" title="Topological space">topology</a> is the space of <a href="Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a>.

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

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

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

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</math><img src="./d2c581e62b55945f57893cfa9e5f5f11b3c84a81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.716ex; height:2.176ex;" alt="{\displaystyle \Psi :\mathbb {R} \to \mathbb {R} }" loading="lazy"> given by

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<annotation encoding="application/x-tex">{\displaystyle \Psi (x)={\begin{cases}\exp \left({\frac {1}{x^{2}-1}}\right),&{\text{ if }}|x|<1,\\0,&{\text{ if }}|x|\geq 1,\end{cases}}}</annotation>

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is an example of a bump function in one dimension. Note that the support of this function is the closed interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]}">

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

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</math><img src="./51e3b7f14a6f70e614728c583409a0b9a8b9de01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle [-1,1]}" loading="lazy">. In fact, by definition of <a href="Support_(mathematics)" title="Support (mathematics)">support</a>, we have that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {supp} (\Psi ):={\overline {\{x\in \mathbb {R} :\Psi (x)\neq 0\}}}={\overline {(-1,1)}}}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {supp} (\Psi ):={\overline {\{x\in \mathbb {R} :\Psi (x)\neq 0\}}}={\overline {(-1,1)}}}</annotation>

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</math><img src="./aad7dc7ef32ceb9e0e08aa29e0a741bebd8235cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.78ex; height:3.676ex;" alt="{\displaystyle \operatorname {supp} (\Psi ):={\overline {\{x\in \mathbb {R} :\Psi (x)\neq 0\}}}={\overline {(-1,1)}}}" loading="lazy">, where the closure is taken with respect the Euclidean topology of the real line. The proof of smoothness follows along the same lines as for the related function discussed in the <a href="Non-analytic_smooth_function" title="Non-analytic smooth function">Non-analytic smooth function</a> article. This function can be interpreted as the <a href="Gaussian_function" title="Gaussian function">Gaussian function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp \left(-y^{2}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle \exp \left(-y^{2}\right)}</annotation>

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</math><img src="./c80f5d035b28c020de22ae9798db8fc4770b7736.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.705ex; height:3.343ex;" alt="{\displaystyle \exp \left(-y^{2}\right)}" loading="lazy"> scaled to fit into the unit disc: the substitution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}={1}/{\left(1-x^{2}\right)}}">

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<annotation encoding="application/x-tex">{\displaystyle y^{2}={1}/{\left(1-x^{2}\right)}}</annotation>

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</math><img src="./d9c68fbf3c69ccf5bd92c30cbb8741b8ef6dbba9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.155ex; height:3.343ex;" alt="{\displaystyle y^{2}={1}/{\left(1-x^{2}\right)}}" loading="lazy"> corresponds to sending <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\pm 1}">

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

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</math><img src="./ae75026eda1ba1ac961f5003150288e9d2f3049b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.399ex; height:2.176ex;" alt="{\displaystyle x=\pm 1}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\infty .}">

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

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</math><img src="./d41dfa1aa4a75080507cf4617726153a0bd9da4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.225ex; height:2.009ex;" alt="{\displaystyle y=\infty .}" loading="lazy">

A simple example of a (square) bump function in <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="./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"> variables is obtained by taking the product of <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="./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"> copies of the above bump function in one variable, so

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<annotation encoding="application/x-tex">{\displaystyle \Phi (x_{1},x_{2},\dots ,x_{n})=\Psi (x_{1})\Psi (x_{2})\cdots \Psi (x_{n}).}</annotation>

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A radially symmetric bump function in <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="./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"> variables can be formed by taking the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{n}:\mathbb {R} ^{n}\to \mathbb {R} }">

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

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</math><img src="./a8ee585da3e6937dca86dc887f7506f117e31e43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.152ex; height:2.676ex;" alt="{\displaystyle \Psi _{n}:\mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{n}(\mathbf {x} )=\Psi (|\mathbf {x} |)}">

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{n}(\mathbf {x} )=\Psi (|\mathbf {x} |)}</annotation>

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</math><img src="./bbdc33a007f65eefcb9abda7ec9023507f29b300.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.667ex; height:2.843ex;" alt="{\displaystyle \Psi _{n}(\mathbf {x} )=\Psi (|\mathbf {x} |)}" loading="lazy">. This function is supported on the unit ball centered at the origin.

For another example, take an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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

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

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</math><img src="./3ee6b94416101a5232787c5d1f20362c35adfd49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.066ex; height:2.843ex;" alt="{\displaystyle (c,d)}" loading="lazy"> and zero elsewhere, for example

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<annotation encoding="application/x-tex">{\displaystyle h(x)={\begin{cases}\exp \left(-{\frac {1}{(x-c)(d-x)}}\right),&c<x<d\\0,&\mathrm {otherwise} \end{cases}}}</annotation>

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</math><img src="./c9c7976d579fcc4785d694e35c7be798e7079689.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.46ex; height:7.509ex;" alt="{\displaystyle h(x)={\begin{cases}\exp \left(-{\frac {1}{(x-c)(d-x)}}\right),&c<x<d\\0,&\mathrm {otherwise} \end{cases}}}" loading="lazy">.</dd></dl>

Smooth transition functions

Consider the function

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

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</math><img src="./f44a0b0073f9c132d735fbdb2d25283158564868.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.704ex; height:6.509ex;" alt="{\displaystyle f(x)={\begin{cases}e^{-{\frac {1}{x}}}&{\text{if }}x>0,\\0&{\text{if }}x\leq 0,\end{cases}}}" loading="lazy"></dd></dl>

defined for every <a href="Real_number" title="Real number">real number</a> x.

The function

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<annotation encoding="application/x-tex">{\displaystyle g(x)={\frac {f(x)}{f(x)+f(1-x)}},\qquad x\in \mathbb {R} ,}</annotation>

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</math><img src="./3225b2f3736bae66ae04d18274e3e5ad2d015102.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.042ex; height:6.509ex;" alt="{\displaystyle g(x)={\frac {f(x)}{f(x)+f(1-x)}},\qquad x\in \mathbb {R} ,}" loading="lazy"></dd></dl>

has a strictly positive denominator everywhere on the real line, hence g is also smooth. Furthermore, g(x) = 0 for x ≤ 0 and g(x) = 1 for x ≥ 1, hence it provides a smooth transition from the level 0 to the level 1 in the <a href="Unit_interval" title="Unit interval">unit interval</a> [0, 1]. To have the smooth transition in the real interval [a, b] with a < b, consider the function

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \ni x\mapsto g{\Bigl (}{\frac {x-a}{b-a}}{\Bigr )}.}</annotation>

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</math><img src="./dc23f9215972a64af5076900200b7e2525d94aaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.237ex; height:5.343ex;" alt="{\displaystyle \mathbb {R} \ni x\mapsto g{\Bigl (}{\frac {x-a}{b-a}}{\Bigr )}.}" loading="lazy"></dd></dl>

For real numbers a < b < c < d, the smooth function

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \ni x\mapsto g{\Bigl (}{\frac {x-a}{b-a}}{\Bigr )}\,g{\Bigl (}{\frac {d-x}{d-c}}{\Bigr )}}</annotation>

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</math><img src="./ccb7335b37bdd1d2d441ccec212a1332bd7417b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:30.091ex; height:5.676ex;" alt="{\displaystyle \mathbb {R} \ni x\mapsto g{\Bigl (}{\frac {x-a}{b-a}}{\Bigr )}\,g{\Bigl (}{\frac {d-x}{d-c}}{\Bigr )}}" loading="lazy"></dd></dl>

equals 1 on the closed interval [b, c] and vanishes outside the open interval (a, d), hence it can serve as a bump function.

Caution must be taken since, as example, taking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a=-1\}<\{b=c=0\}<\{d=1\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{a=-1\}<\{b=c=0\}<\{d=1\}}</annotation>

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</math><img src="./0075b157645dc080e8cfd676458510cb4fc0e42f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.311ex; height:2.843ex;" alt="{\displaystyle \{a=-1\}<\{b=c=0\}<\{d=1\}}" loading="lazy">, leads to:

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<annotation encoding="application/x-tex">{\displaystyle q(x)={\frac {1}{1+e^{\frac {1-2|x|}{x^{2}-|x|}}}}}</annotation>

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</math><img src="./5d12ed56a06908ee14ac2637ef22226baeb450a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:18.406ex; height:8.009ex;" alt="{\displaystyle q(x)={\frac {1}{1+e^{\frac {1-2|x|}{x^{2}-|x|}}}}}" loading="lazy"></dd></dl>

which is not an infinitely <a href="Differentiable_function" title="Differentiable function">differentiable function</a> (so, is not "smooth"), so the constraints a < b < c < d must be strictly fulfilled.

Some interesting facts about the function:

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<annotation encoding="application/x-tex">{\displaystyle q(x,a)={\frac {1}{1+e^{\frac {a(1-2|x|)}{x^{2}-|x|}}}}}</annotation>

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</math><img src="./9a19c2324f45482c626a88b034b969b488262e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:22.185ex; height:8.009ex;" alt="{\displaystyle q(x,a)={\frac {1}{1+e^{\frac {a(1-2|x|)}{x^{2}-|x|}}}}}" loading="lazy"></dd></dl>

Are that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\left(x,{\frac {\sqrt {3}}{2}}\right)}">

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<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\left(x,{\frac {\sqrt {3}}{2}}\right)}</annotation>

</semantics>

</math><img src="./2b41f3c1f94990919f805bb67145248b91d18df5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.176ex; height:6.509ex;" alt="{\displaystyle q\left(x,{\frac {\sqrt {3}}{2}}\right)}" loading="lazy"> make smooth transition curves with "almost" constant slope edges (a bump function with true straight slopes is portrayed this <a href="Delay_differential_equation" title="Delay differential equation">Another example</a>).

A proper example of a smooth Bump function would be:

<mrow>

<mtr>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<msup>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mfrac>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mtext>otherwise</mtext>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c9b8b76c0e7eb35b0d4124e714328526a75588aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.345ex; margin-bottom: -0.327ex; width:30.454ex; height:12.509ex;" alt="{\displaystyle u(x)={\begin{cases}1,{\text{if }}x=0,\\0,{\text{if }}|x|\geq 1,\\{\frac {1}{1+e^{\frac {1-2|x|}{x^{2}-|x|}}}},{\text{otherwise}},\end{cases}}}" loading="lazy"></dd></dl>

A proper example of a smooth transition function will be:

<mrow>

<mtr>

<mtd>

<mfrac>

<mrow>

<msup>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

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</msup>

</mrow>

</mfrac>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f298b6465d07268ef5edeccd3e553a95f811c639.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:33.548ex; height:11.843ex;" alt="{\displaystyle w(x)={\begin{cases}{\frac {1}{1+e^{\frac {2x-1}{x^{2}-x}}}}&{\text{if }}0<x<1,\\0&{\text{if }}x\leq 0,\\1&{\text{if }}x\geq 1,\end{cases}}}" loading="lazy"></dd></dl>

where could be noticed that it can be represented also through <a href="Hyperbolic_functions" title="Hyperbolic functions">Hyperbolic functions</a>:

<mfrac>

<mrow>

<msup>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

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</mfrac>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1+e^{\frac {2x-1}{x^{2}-x}}}}={\frac {1}{2}}\left(1-\tanh \left({\frac {2x-1}{2(x^{2}-x)}}\right)\right)}</annotation>

</semantics>

</math><img src="./77d6afd5bd16d97def485315c273738f59d66daa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:41.7ex; height:7.676ex;" alt="{\displaystyle {\frac {1}{1+e^{\frac {2x-1}{x^{2}-x}}}}={\frac {1}{2}}\left(1-\tanh \left({\frac {2x-1}{2(x^{2}-x)}}\right)\right)}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Existence_of_bump_functions">Existence of bump functions</h2></div>

It is possible to construct bump functions "to specifications". Stated formally, if <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"> is an arbitrary <a href="Compact_set" class="mw-redirect" title="Compact set">compact set</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> dimensions and <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"> is an <a href="Open_set" title="Open set">open set</a> containing <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="./9fea4b5006afa0eb81c21b577efe304ab545d571.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle K,}" loading="lazy"> there exists a bump function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"> which is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<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"> on <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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> outside 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="./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"> Since <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"> can be taken to be a very small neighborhood of <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="./9fea4b5006afa0eb81c21b577efe304ab545d571.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle K,}" loading="lazy"> this amounts to being able to construct a function that is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<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"> on <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"> and falls off rapidly to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> outside of <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="./9fea4b5006afa0eb81c21b577efe304ab545d571.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle K,}" loading="lazy"> while still being smooth.

Bump functions defined in terms of convolution

The construction proceeds as follows. One considers a compact neighborhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

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

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K\subseteq V^{\circ }\subseteq V\subseteq U.}</annotation>

</semantics>

</math><img src="./5e2c4e2f7153b49233f87fb5c7ec305b202cfa3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.549ex; height:2.509ex;" alt="{\displaystyle K\subseteq V^{\circ }\subseteq V\subseteq U.}" loading="lazy"> The <a href="Indicator_function" title="Indicator function">characteristic function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{V}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./592e19fdc1d0855754dc7ee061f043263edd9260.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.951ex; height:2.009ex;" alt="{\displaystyle \chi _{V}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> will be equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> outside of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ace9595e3ce66fdec7e9d30202626accd676b11e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.434ex; height:2.509ex;" alt="{\displaystyle V,}" loading="lazy"> so in particular, it will be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<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"> on <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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> outside 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="./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"> This function is not smooth however. The key idea is to smooth <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{V}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./592e19fdc1d0855754dc7ee061f043263edd9260.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.951ex; height:2.009ex;" alt="{\displaystyle \chi _{V}}" loading="lazy"> a bit, by taking the <a href="Convolution" title="Convolution">convolution</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{V}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./592e19fdc1d0855754dc7ee061f043263edd9260.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.951ex; height:2.009ex;" alt="{\displaystyle \chi _{V}}" loading="lazy"> with a <a href="Mollifier" title="Mollifier">mollifier</a>. The latter is just a bump function with a very small support and whose integral is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./af8c4e445819b13a052647aa3eb2be990b0a4b24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 1.}" loading="lazy"> Such a mollifier can be obtained, for example, by taking the bump function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> from the previous section and performing appropriate scalings.

Bump functions defined in terms of a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c:\mathbb {R} \to [0,\infty )}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c40073d19842bc5fc4c1ee74aabfa6087517b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.308ex; height:2.843ex;" alt="{\displaystyle c:\mathbb {R} \to [0,\infty )}" loading="lazy"> with support <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,0]}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0241c015ef4c611c9c9aeafb395e7c4a16178405.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle (-\infty ,0]}" loading="lazy">

An alternative construction that does not involve convolution is now detailed.

It begins by constructing a smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./306c097f43c91dce633d12cde024948d39e73752.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.404ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"> that is positive on a given open subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subseteq \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1a4ffabb41c43db92055a8fd5be74cd7633756a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.778ex; height:2.509ex;" alt="{\displaystyle U\subseteq \mathbb {R} ^{n}}" loading="lazy"> and vanishes off 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="./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-FOOTNOTENestruev202013–16-1">[1]</a> This function's support is equal to the closure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {U}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./19c0e1c90cfd1f74816d2ac331fe88656cfe260d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:3.009ex;" alt="{\displaystyle {\overline {U}}}" loading="lazy"> 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"> in <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="./d7035fcb9fe3ebecc6bc9f372f82d0352202c8bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{n},}" loading="lazy"> so if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {U}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./19c0e1c90cfd1f74816d2ac331fe88656cfe260d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:3.009ex;" alt="{\displaystyle {\overline {U}}}" loading="lazy"> is compact, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

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

Start with any smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c:\mathbb {R} \to \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1286fd7708e9874fa113b94e4da3a2eb0715f657.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.914ex; height:2.176ex;" alt="{\displaystyle c:\mathbb {R} \to \mathbb {R} }" loading="lazy"> that vanishes on the negative reals and is positive on the positive reals (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d9ee918699d0cb4b8c633cc1f520a8a7a174f44a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c=0}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,0)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8399275ffe1d565a9df43ae1abd4b3712ad94527.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.138ex; height:2.843ex;" alt="{\displaystyle (-\infty ,0)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c>0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2ba126f626d61752f62eaacaf11761a54de4dc84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c>0}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,\infty ),}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d2775077631b691fb6b7132ce5eb139ee209941f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (0,\infty ),}" loading="lazy"> where continuity from the left necessitates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(0)=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dd3b2e94eaaee7cd2fb59c8435c3cb39c07c0420.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.24ex; height:2.843ex;" alt="{\displaystyle c(0)=0}" loading="lazy">); an example of such a function is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(x):=e^{-1/x}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c(x):=e^{-1/x}}</annotation>

</semantics>

</math><img src="./b17e534092a8fca634651695ee53b1f01e34d95b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.07ex; height:3.343ex;" alt="{\displaystyle c(x):=e^{-1/x}}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./80d24be5f0eb4a9173da6038badc8659546021d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(x):=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./19b5e58537bc2e53198da3b8a0fcb7884b2354e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.054ex; height:2.843ex;" alt="{\displaystyle c(x):=0}" loading="lazy"> otherwise.<a href="#cite_note-FOOTNOTENestruev202013–16-1">[1]</a>

Fix an open subset <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"> and denote the usual <a href="Euclidean_norm" class="mw-redirect" title="Euclidean norm">Euclidean norm</a> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./113f0d8fe6108fc1c5e9802f7c3f634f5480b3d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.004ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|}" loading="lazy"> (so <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 endowed with the usual <a href="Euclidean_metric" class="mw-redirect" title="Euclidean metric">Euclidean metric</a>).

The following construction defines a smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./306c097f43c91dce633d12cde024948d39e73752.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.404ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"> that is positive 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"> and vanishes outside 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="./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-FOOTNOTENestruev202013–16-1">[1]</a> So in particular, if <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"> is relatively compact then this 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"> will be a bump function.

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

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c964e8bbffc01889801b4aaa7327cad993524ff1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.778ex; height:2.343ex;" alt="{\displaystyle U=\mathbb {R} ^{n}}" loading="lazy"> then let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./60eab5895ca5fe74555e9d59e499c94e1d2ef3f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle f=1}" loading="lazy"> while if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=\varnothing }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./51221c4e094c4e8a284558ec479beff7924160b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.689ex; height:2.176ex;" alt="{\displaystyle U=\varnothing }" loading="lazy"> then let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1ee0fdf0f50fcba5afe3e856fcc7dc6acfa61014.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle f=0}" loading="lazy">; so assume <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"> is neither of these. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(U_{k}\right)_{k=1}^{\infty }}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(U_{k}\right)_{k=1}^{\infty }}</annotation>

</semantics>

</math><img src="./fd802b838eec7f459b72a5bddec125a6b5a1b0eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.675ex; height:3.176ex;" alt="{\displaystyle \left(U_{k}\right)_{k=1}^{\infty }}" loading="lazy"> be an <a href="Cover_(topology)" title="Cover (topology)">open cover</a> 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"> by open balls where the open ball <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./924cf85ae48192ae1c1ce15c109032e0a69b1014.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.676ex; height:2.509ex;" alt="{\displaystyle U_{k}}" loading="lazy"> has radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{k}>0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ad87d3ef2fd138d184882074dbcfdaaae102cb14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.398ex; height:2.509ex;" alt="{\displaystyle r_{k}>0}" loading="lazy"> and center <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k}\in U.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a6040f7b983915914708a7fa896e091db08bddb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.589ex; height:2.509ex;" alt="{\displaystyle a_{k}\in U.}" loading="lazy"> Then the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{k}:\mathbb {R} ^{n}\to \mathbb {R} }">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./584ec18a8afecff0fc7d816b086a894c96452e06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.354ex; height:2.676ex;" alt="{\displaystyle f_{k}:\mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{k}(x)=c\left(r_{k}^{2}-\left\|x-a_{k}\right\|^{2}\right)}">

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

<msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{k}(x)=c\left(r_{k}^{2}-\left\|x-a_{k}\right\|^{2}\right)}</annotation>

</semantics>

</math><img src="./e9dc70033e0d9515cfca87e3ac24345bf6003138.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.48ex; height:4.843ex;" alt="{\displaystyle f_{k}(x)=c\left(r_{k}^{2}-\left\|x-a_{k}\right\|^{2}\right)}" loading="lazy"> is a smooth function that is positive on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./924cf85ae48192ae1c1ce15c109032e0a69b1014.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.676ex; height:2.509ex;" alt="{\displaystyle U_{k}}" loading="lazy"> and vanishes off of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{k}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0f7786cefa9d767815d2d03d5574824071fae75f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.323ex; height:2.509ex;" alt="{\displaystyle U_{k}.}" loading="lazy"><a href="#cite_note-FOOTNOTENestruev202013–16-1">[1]</a>

For every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} ,}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./41f2d6e44ab029b7cee559d32df7149e9e221baf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.377ex; height:2.509ex;" alt="{\displaystyle k\in \mathbb {N} ,}" loading="lazy"> let

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mtext> satisfy </mtext>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{k}=\sup \left\{\left|{\frac {\partial ^{p}f_{k}}{\partial ^{p_{1}}x_{1}\cdots \partial ^{p_{n}}x_{n}}}(x)\right|~:~x\in \mathbb {R} ^{n}{\text{ and }}p_{1},\ldots ,p_{n}\in \mathbb {Z} {\text{ satisfy }}0\leq p_{i}\leq k{\text{ and }}p=\sum _{i}p_{i}\right\},}</annotation>

</semantics>

</math>

where this <a href="Supremum" class="mw-redirect" title="Supremum">supremum</a> is not equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./bddbb0e4420a7e744cf71bd71216e11b0bf88831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }" loading="lazy"> (so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./43175c381ee6ba2694520af1f1a26c676a2726ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.343ex; height:2.509ex;" alt="{\displaystyle M_{k}}" loading="lazy"> is a non-negative real number) because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbb {R} ^{n}\setminus U_{k}\right)\cup {\overline {U_{k}}}=\mathbb {R} ^{n},}">

<mrow>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mover>

<msub>

</mrow>

</msub>

</mover>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(\mathbb {R} ^{n}\setminus U_{k}\right)\cup {\overline {U_{k}}}=\mathbb {R} ^{n},}</annotation>

</semantics>

</math><img src="./9563f4a01e3e4adbb6f9634a1ae9046f3613f27d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.592ex; height:3.509ex;" alt="{\displaystyle \left(\mathbb {R} ^{n}\setminus U_{k}\right)\cup {\overline {U_{k}}}=\mathbb {R} ^{n},}" loading="lazy"> the partial derivatives all vanish (equal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">) at any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

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

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d7173866a3fa5e80ca94f4715cb6383802bd0a0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.323ex; height:2.509ex;" alt="{\displaystyle U_{k},}" loading="lazy"> while on the compact set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {U_{k}}},}">

<mover>

<msub>

</mrow>

</msub>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f750d151cec12803fb00048621c81860ee5483e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.438ex; height:3.343ex;" alt="{\displaystyle {\overline {U_{k}}},}" loading="lazy"> the values of each of the (finitely many) partial derivatives are (uniformly) bounded above by some non-negative real number.<a href="#cite_note-2">[note 1]</a>

The series

</mrow>

</mrow>

</munderover>

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f~:=~\sum _{k=1}^{\infty }{\frac {f_{k}}{2^{k}M_{k}}}}</annotation>

</semantics>

</math>

converges uniformly on <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"> to a smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./306c097f43c91dce633d12cde024948d39e73752.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.404ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"> that is positive 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"> and vanishes off 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="./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-FOOTNOTENestruev202013–16-1">[1]</a>

Moreover, for any non-negative integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1},\ldots ,p_{n}\in \mathbb {Z} ,}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{1},\ldots ,p_{n}\in \mathbb {Z} ,}</annotation>

</semantics>

</math><img src="./1ba3eba357ac6b0972deec4e4504a823d79c33f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:14.917ex; height:2.509ex;" alt="{\displaystyle p_{1},\ldots ,p_{n}\in \mathbb {Z} ,}" loading="lazy"><a href="#cite_note-FOOTNOTENestruev202013–16-1">[1]</a>

<mfrac>

<msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{p_{1}+\cdots +p_{n}}}{\partial ^{p_{1}}x_{1}\cdots \partial ^{p_{n}}x_{n}}}f~=~\sum _{k=1}^{\infty }{\frac {1}{2^{k}M_{k}}}{\frac {\partial ^{p_{1}+\cdots +p_{n}}f_{k}}{\partial ^{p_{1}}x_{1}\cdots \partial ^{p_{n}}x_{n}}}}</annotation>

</semantics>

</math>

where this series also converges uniformly on <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"> (because whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq p_{1}+\cdots +p_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\geq p_{1}+\cdots +p_{n}}</annotation>

</semantics>

</math><img src="./f7d6eb19ba24e23444bf3a92aae9854ed5507bd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.325ex; height:2.509ex;" alt="{\displaystyle k\geq p_{1}+\cdots +p_{n}}" loading="lazy"> then 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 term's <a href="Absolute_value" title="Absolute value">absolute value</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq {\tfrac {M_{k}}{2^{k}M_{k}}}={\tfrac {1}{2^{k}}}}">

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mstyle>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \leq {\tfrac {M_{k}}{2^{k}M_{k}}}={\tfrac {1}{2^{k}}}}</annotation>

</semantics>

</math><img src="./aaf8af2150c7ce44aa946bc1c93a29c99e3df50b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.041ex; height:4.843ex;" alt="{\displaystyle \leq {\tfrac {M_{k}}{2^{k}M_{k}}}={\tfrac {1}{2^{k}}}}" loading="lazy">). This completes the construction.

As a corollary, given two disjoint closed subsets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./96c3298ea9aa77c226be56a7d8515baaa517b90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.541ex; height:2.509ex;" alt="{\displaystyle A,B}" 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="./d7035fcb9fe3ebecc6bc9f372f82d0352202c8bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{n},}" loading="lazy"> the above construction guarantees the existence of smooth non-negative functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{A},f_{B}:\mathbb {R} ^{n}\to [0,\infty )}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{A},f_{B}:\mathbb {R} ^{n}\to [0,\infty )}</annotation>

</semantics>

</math><img src="./086786aa9ec278e2801e5192873cfb76615b0a97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.776ex; height:2.843ex;" alt="{\displaystyle f_{A},f_{B}:\mathbb {R} ^{n}\to [0,\infty )}" loading="lazy"> such that for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} ^{n},}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6687adfceb0ccfd4fbbb619cf4768712691fa37f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.714ex; height:2.676ex;" alt="{\displaystyle x\in \mathbb {R} ^{n},}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{A}(x)=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d85dd48543b62a3e329f90ff2382a0b020c7845b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.004ex; height:2.843ex;" alt="{\displaystyle f_{A}(x)=0}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in A,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7c649021f8a56d0234bfa3eae952bcab3edad556.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.56ex; height:2.509ex;" alt="{\displaystyle x\in A,}" loading="lazy"> and similarly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{B}(x)=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ec176e9945016932594226f7d96ab583643a1bdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.019ex; height:2.843ex;" alt="{\displaystyle f_{B}(x)=0}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in B,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aa38c705b94de426e7d467ab8f57e86941d1b545.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.581ex; height:2.509ex;" alt="{\displaystyle x\in B,}" loading="lazy"> then the function

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h~:=~{\frac {f_{A}}{f_{A}+f_{B}}}:\mathbb {R} ^{n}\to [0,1]}</annotation>

</semantics>

</math>

is smooth and for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} ^{n},}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6687adfceb0ccfd4fbbb619cf4768712691fa37f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.714ex; height:2.676ex;" alt="{\displaystyle x\in \mathbb {R} ^{n},}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x)=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b13fa03ef9ad62ec2199ec214eb3af674ef391e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.739ex; height:2.843ex;" alt="{\displaystyle h(x)=0}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in A,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7c649021f8a56d0234bfa3eae952bcab3edad556.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.56ex; height:2.509ex;" alt="{\displaystyle x\in A,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x)=1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6397dc3ddab50b85f5c07df2d057a05b65442572.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.739ex; height:2.843ex;" alt="{\displaystyle h(x)=1}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in B,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aa38c705b94de426e7d467ab8f57e86941d1b545.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.581ex; height:2.509ex;" alt="{\displaystyle x\in B,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<h(x)<1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aa9d8f014ba3e65ebcb80c7871d4e3bbce176144.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13ex; height:2.843ex;" alt="{\displaystyle 0<h(x)<1}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\not \in A\cup B.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\not \in A\cup B.}</annotation>

</semantics>

</math><img src="./5971153754da9d05ad21da94741b98f3c3b39991.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.907ex; height:2.676ex;" alt="{\displaystyle x\not \in A\cup B.}" loading="lazy"><a href="#cite_note-FOOTNOTENestruev202013–16-1">[1]</a>

In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x)\neq 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./986c6b3b7a1f9bc97e81e86c1b71209c35308518.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.739ex; height:2.843ex;" alt="{\displaystyle h(x)\neq 0}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} ^{n}\smallsetminus A,}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3e4cfc1bed77df93220b5d8799e23c7bfdc9e470.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.297ex; height:2.676ex;" alt="{\displaystyle x\in \mathbb {R} ^{n}\smallsetminus A,}" loading="lazy"> so if in addition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U:=\mathbb {R} ^{n}\smallsetminus A}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./53986cc3a584126bb9818a07e211ffd41e1d4c06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.008ex; height:2.343ex;" alt="{\displaystyle U:=\mathbb {R} ^{n}\smallsetminus A}" loading="lazy"> is relatively compact in <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"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cap B=\varnothing }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\cap B=\varnothing }</annotation>

</semantics>

</math><img src="./1325e582d7bc7ad1e30dfef4ee67747895af068e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.996ex; height:2.176ex;" alt="{\displaystyle A\cap B=\varnothing }" loading="lazy"> implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subseteq U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5957e2cd0dc4c3128bff9d5465b45489dc5b10a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.645ex; height:2.343ex;" alt="{\displaystyle B\subseteq U}" loading="lazy">) then <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"> will be a smooth bump function with support in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {U}}.}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9e0ee762579e814729b63d34ca8c5d06d7e42be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.671ex; height:3.009ex;" alt="{\displaystyle {\overline {U}}.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Properties_and_uses">Properties and uses</h2></div>

While bump functions are smooth, the <a href="Identity_theorem" title="Identity theorem">identity theorem</a> prohibits their being <a href="Analytic_function" title="Analytic function">analytic</a> unless they <a href="Zero_of_a_function" title="Zero of a function">vanish</a> identically. Bump functions are often used as <a href="Mollifier" title="Mollifier">mollifiers</a>, as smooth <a href="Cutoff_function" class="mw-redirect" title="Cutoff function">cutoff functions</a>, and to form smooth <a href="Partitions_of_unity" class="mw-redirect" title="Partitions of unity">partitions of unity</a>. They are the most common class of <a href="Test_functions" class="mw-redirect" title="Test functions">test functions</a> used in analysis. The space of bump functions is closed under many operations. For instance, the sum, product, or <a href="Convolution" title="Convolution">convolution</a> of two bump functions is again a bump function, and any <a href="Differential_operator" title="Differential operator">differential operator</a> with smooth coefficients, when applied to a bump function, will produce another bump function.

If the boundaries of the Bump function domain is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial x,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1a17da165de46b2be9d7a89864c156f59876138b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.295ex; height:2.509ex;" alt="{\displaystyle \partial x,}" loading="lazy"> to fulfill the requirement of "smoothness", it has to preserve the continuity of all its derivatives, which leads to the following requirement at the boundaries of its domain:

<msup>

</mrow>

</msup>

</mrow>

</munder>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{x\to \partial x^{\pm }}{\frac {d^{n}}{dx^{n}}}f(x)=0,\,{\text{ for all }}n\geq 0,\,n\in \mathbb {Z} }</annotation>

</semantics>

</math>

The <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of a bump function is a (real) analytic function, and it can be extended to the whole <a href="Complex_plane" title="Complex plane">complex plane</a>: hence it cannot be compactly supported unless it is zero, since the only entire analytic bump function is the zero function (see <a href="Paley%E2%80%93Wiener_theorem" title="Paley–Wiener theorem">Paley–Wiener theorem</a> and <a href="Liouville's_theorem_(complex_analysis)" title="Liouville's theorem (complex analysis)">Liouville's theorem</a>). Because the bump function is infinitely differentiable, its Fourier transform must decay faster than any finite power of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/k}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a7e9fedad8c70c6331b2640b56c23cef8c884e1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.536ex; height:2.843ex;" alt="{\displaystyle 1/k}" loading="lazy"> for a large angular frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |k|.}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6ad1bbd8e71eb558aa1689b42ba7786c1b72a84a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.152ex; height:2.843ex;" alt="{\displaystyle |k|.}" loading="lazy"><a href="#cite_note-3">[2]</a> The Fourier transform of the particular bump function

<msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi (x)=e^{-1/(1-x^{2})}\mathbf {1} _{\{|x|<1\}}}</annotation>

</semantics>

</math>

from above can be analyzed by a <a href="Saddle-point_method" class="mw-redirect" title="Saddle-point method">saddle-point method</a>, and decays asymptotically as

</mrow>

<msup>

</mrow>

</mrow>

</mrow>

</msup>

<msup>

<msqrt>

</mrow>

</mrow>

</msqrt>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |k|^{-3/4}e^{-{\sqrt {|k|}}}}</annotation>

</semantics>

</math>

for large <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |k|.}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6ad1bbd8e71eb558aa1689b42ba7786c1b72a84a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.152ex; height:2.843ex;" alt="{\displaystyle |k|.}" loading="lazy"><a href="#cite_note-4">[3]</a>

<ul><li><a href="Cutoff_function" class="mw-redirect" title="Cutoff function">Cutoff function</a> – Integration kernels for smoothing out sharp featuresPages displaying short descriptions of redirect targets</li>

<li><a href="Laplacian_of_the_indicator" title="Laplacian of the indicator">Laplacian of the indicator</a> – Limit of sequence of smooth functions</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="Schwartz_space" title="Schwartz space">Schwartz space</a> – Function space of all functions whose derivatives are rapidly decreasing</li></ul>

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

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<li id="cite_note-2"><a href="#cite_ref-2">^</a> The partial derivatives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{p}f_{k}}{\partial ^{p_{1}}x_{1}\cdots \partial ^{p_{n}}x_{n}}}:\mathbb {R} ^{n}\to \mathbb {R} }">

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{p}f_{k}}{\partial ^{p_{1}}x_{1}\cdots \partial ^{p_{n}}x_{n}}}:\mathbb {R} ^{n}\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./038046314c9ed8efa087ce7904fbd4b8808719ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.993ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{p}f_{k}}{\partial ^{p_{1}}x_{1}\cdots \partial ^{p_{n}}x_{n}}}:\mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"> are continuous functions so the image of the compact subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {U_{k}}}}">

<mover>

<msub>

</mrow>

</msub>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3d91465a6970549880550a927f206f0c99b5c9b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.791ex; height:3.343ex;" alt="{\displaystyle {\overline {U_{k}}}}" loading="lazy"> is a compact subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} .}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dc9de9049e03e5e5a0cab57076dbe4a369c1e3a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} .}" loading="lazy"> The supremum is over all non-negative integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq p=p_{1}+\cdots +p_{n}\leq k}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\leq p=p_{1}+\cdots +p_{n}\leq k}</annotation>

</semantics>

</math><img src="./aff25d5c8d354056e61a2cedf76b4515068cb90f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.854ex; height:2.509ex;" alt="{\displaystyle 0\leq p=p_{1}+\cdots +p_{n}\leq k}" loading="lazy"> where because <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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> are fixed, this supremum is taken over only finitely many partial derivatives, which is why <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{k}<\infty .}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8d555ab3d67d981d6397bf5d49f9494d6e912813.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.412ex; height:2.509ex;" alt="{\displaystyle M_{k}<\infty .}" loading="lazy">

</li>

</ol></div></div>

<li id="cite_note-FOOTNOTENestruev202013–16-1">^ <a href="#cite_ref-FOOTNOTENestruev202013–16_1-0">a</a> <a href="#cite_ref-FOOTNOTENestruev202013–16_1-1">b</a> <a href="#cite_ref-FOOTNOTENestruev202013–16_1-2">c</a> <a href="#cite_ref-FOOTNOTENestruev202013–16_1-3">d</a> <a href="#cite_ref-FOOTNOTENestruev202013–16_1-4">e</a> <a href="#cite_ref-FOOTNOTENestruev202013–16_1-5">f</a> <a href="#cite_ref-FOOTNOTENestruev202013–16_1-6">g</a> <a href="#CITEREFNestruev2020">Nestruev 2020</a>, pp. 13–16.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> K. O. Mead and L. M. Delves, "On the convergence rate of generalized Fourier expansions," IMA J. Appl. Math., vol. 12, pp. 247–259 (1973) <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fimamat%2F12.3.247">10.1093/imamat/12.3.247</a>.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="Steven_G._Johnson" title="Steven G. Johnson">Steven G. Johnson</a>, <a rel="nofollow" class="external text" href="https://arxiv.org/abs/1508.04376">Saddle-point integration of C∞ "bump" functions</a>, arXiv:1508.04376 (2015).

</li>

</ol></div></div>

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

<ul><li><cite id="CITEREFNestruev2020" class="citation book cs1">Nestruev, Jet (10 September 2020). Smooth Manifolds and Observables. <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>. Vol. 220. Cham, Switzerland: <a href="Springer_Nature" title="Springer Nature">Springer Nature</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-030-45649-8</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1195920718">1195920718</a>.</cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-06-09" href="https://en.wikipedia.org/wiki/?title=Bump_function&oldid=1294793124">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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