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<title>Non-analytic smooth function</title>

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Non-analytic smooth function

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In <a href="Mathematics" title="Mathematics">mathematics</a>, <a href="Smooth_function" class="mw-redirect" title="Smooth function">smooth functions</a> (also called infinitely <a href="Differentiable_function" title="Differentiable function">differentiable</a> functions) and <a href="Analytic_function" title="Analytic function">analytic functions</a> are two very important types of <a href="Function_(mathematics)" title="Function (mathematics)">functions</a>. One can easily prove that any analytic function of a <a href="Real_number" title="Real number">real</a> <a href="Argument_of_a_function" title="Argument of a function">argument</a> is smooth. The <a href="Converse_(logic)" title="Converse (logic)">converse</a> is not true, as demonstrated with the <a href="Counterexample" title="Counterexample">counterexample</a> below.

One of the most important applications of smooth functions with <a href="Compact_support" class="mw-redirect" title="Compact support">compact support</a> is the construction of so-called <a href="Mollifier" title="Mollifier">mollifiers</a>, which are important in theories of <a href="Generalized_function" title="Generalized function">generalized functions</a>, such as <a href="Laurent_Schwartz" title="Laurent Schwartz">Laurent Schwartz</a>'s theory of <a href="Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a>.

The existence of smooth but non-analytic functions represents one of the main differences between <a href="Differential_geometry" title="Differential geometry">differential geometry</a> and <a href="Complex_manifold" title="Complex manifold">analytic geometry</a>. In terms of <a href="Sheaf_theory" class="mw-redirect" title="Sheaf theory">sheaf theory</a>, this difference can be stated as follows: the sheaf of differentiable functions on a <a href="Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a> is <a href="Fine_sheaf" class="mw-redirect" title="Fine sheaf">fine</a>, in contrast with the analytic case.

The functions below are generally used to build up <a href="Partition_of_unity" title="Partition of unity">partitions of unity</a> on differentiable manifolds.

<div class="mw-heading mw-heading2"><h2 id="An_example_function">An example function</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Definition_of_the_function">Definition of the function</h3></div>

Consider the function

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\begin{cases}e^{-{\frac {1}{x}}}&{\text{if }}x>0,\\0&{\text{if }}x\leq 0,\end{cases}}}">

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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.

<div class="mw-heading mw-heading3"><h3 id="The_function_is_smooth">The function is smooth</h3></div>

The function f has <a href="Continuous_function" title="Continuous function">continuous</a> <a href="Derivative" title="Derivative">derivatives</a> of all orders at every point x of the <a href="Real_line" class="mw-redirect" title="Real line">real line</a>. The formula for these derivatives is

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{(n)}(x)={\begin{cases}\displaystyle {\frac {p_{n}(x)}{x^{2n}}}\,f(x)&{\text{if }}x>0,\\0&{\text{if }}x\leq 0,\end{cases}}}">

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

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</math><img src="./34c75f89d2aa3003732b97bfbbb39adef53f0129.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:34.54ex; height:8.509ex;" alt="{\displaystyle f^{(n)}(x)={\begin{cases}\displaystyle {\frac {p_{n}(x)}{x^{2n}}}\,f(x)&{\text{if }}x>0,\\0&{\text{if }}x\leq 0,\end{cases}}}" loading="lazy"></dd></dl>

where pn(x) is a <a href="Polynomial" title="Polynomial">polynomial</a> of <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> n − 1 given <a href="Recursion" title="Recursion">recursively</a> by p1(x) = 1 and

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<annotation encoding="application/x-tex">{\displaystyle p_{n+1}(x)=x^{2}p_{n}'(x)-(2nx-1)p_{n}(x)}</annotation>

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</math><img src="./e68fb4cae2577137df5ac4cd4953490a1ced4e76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:36.792ex; height:3.176ex;" alt="{\displaystyle p_{n+1}(x)=x^{2}p_{n}'(x)-(2nx-1)p_{n}(x)}" loading="lazy"></dd></dl>

for any positive <a href="Integer" title="Integer">integer</a> n. From this formula, it is not completely clear that the derivatives are continuous at 0; this follows from the <a href="One-sided_limit" title="One-sided limit">one-sided limit</a>

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<annotation encoding="application/x-tex">{\displaystyle \lim _{x\searrow 0}{\frac {e^{-{\frac {1}{x}}}}{x^{m}}}=0}</annotation>

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</math><img src="./0cb5900177dad3b6b09a50cdba1cffb1c8a2487b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.083ex; height:6.843ex;" alt="{\displaystyle \lim _{x\searrow 0}{\frac {e^{-{\frac {1}{x}}}}{x^{m}}}=0}" loading="lazy"></dd></dl>

for any <a href="Nonnegative" class="mw-redirect" title="Nonnegative">nonnegative</a> integer m.

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<th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">Detailed proof of smoothness</div>

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By the <a href="Exponential_function#Formal_definition" title="Exponential function">power series representation of the exponential function</a>, we have for every <a href="Natural_number" title="Natural number">natural number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> (including zero)

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{x^{m}}}=x{\Bigl (}{\frac {1}{x}}{\Bigr )}^{m+1}\leq (m+1)!\,x\sum _{n=0}^{\infty }{\frac {1}{n!}}{\Bigl (}{\frac {1}{x}}{\Bigr )}^{n}=(m+1)!\,xe^{\frac {1}{x}},\qquad x>0,}</annotation>

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</math><img src="./18d1f680df8a3057571f05b6f03c245b9a4f50e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:71.614ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{x^{m}}}=x{\Bigl (}{\frac {1}{x}}{\Bigr )}^{m+1}\leq (m+1)!\,x\sum _{n=0}^{\infty }{\frac {1}{n!}}{\Bigl (}{\frac {1}{x}}{\Bigr )}^{n}=(m+1)!\,xe^{\frac {1}{x}},\qquad x>0,}" loading="lazy"></dd></dl>

because all the positive terms for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\neq m+1}">

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

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</math><img src="./152ac534aa1d81d9016bc5bb999474a09184eaea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.536ex; height:2.676ex;" alt="{\displaystyle n\neq m+1}" loading="lazy"> are added. Therefore, dividing this inequality by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{\frac {1}{x}}}">

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

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</math><img src="./7a5f1f1dafd6e447280d5b9447fb82dec67bbb8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.915ex; height:3.343ex;" alt="{\displaystyle e^{\frac {1}{x}}}" loading="lazy"> and taking the <a href="One-sided_limit" title="One-sided limit">limit from above</a>,

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<annotation encoding="application/x-tex">{\displaystyle \lim _{x\searrow 0}{\frac {e^{-{\frac {1}{x}}}}{x^{m}}}\leq (m+1)!\lim _{x\searrow 0}x=0.}</annotation>

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</math><img src="./b49c436ac35e84069779e5a3eef48d5d9d66197a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.837ex; height:6.843ex;" alt="{\displaystyle \lim _{x\searrow 0}{\frac {e^{-{\frac {1}{x}}}}{x^{m}}}\leq (m+1)!\lim _{x\searrow 0}x=0.}" loading="lazy"></dd></dl>

We now prove the formula for the nth derivative of f by <a href="Mathematical_induction" title="Mathematical induction">mathematical induction</a>. Using the <a href="Chain_rule" title="Chain rule">chain rule</a>, the <a href="Reciprocal_rule" title="Reciprocal rule">reciprocal rule</a>, and the fact that the derivative of the exponential function is again the exponential function, we see that the formula is correct for the first derivative of f for all x > 0 and that p1(x) is a polynomial of degree 0. Of course, the derivative of f is zero for x < 0.

It remains to show that the right-hand side derivative of f at x = 0 is zero. Using the above limit, we see that

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<annotation encoding="application/x-tex">{\displaystyle f'(0)=\lim _{x\searrow 0}{\frac {f(x)-f(0)}{x-0}}=\lim _{x\searrow 0}{\frac {e^{-{\frac {1}{x}}}}{x}}=0.}</annotation>

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</math><img src="./862951c5b9c124136d7ccaf07abc299567c07335.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.041ex; height:6.843ex;" alt="{\displaystyle f'(0)=\lim _{x\searrow 0}{\frac {f(x)-f(0)}{x-0}}=\lim _{x\searrow 0}{\frac {e^{-{\frac {1}{x}}}}{x}}=0.}" loading="lazy"></dd></dl>

The induction step from n to n + 1 is similar. For x > 0 we get for the derivative

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f^{(n+1)}(x)&={\biggl (}{\frac {p'_{n}(x)}{x^{2n}}}-2n{\frac {p_{n}(x)}{x^{2n+1}}}+{\frac {p_{n}(x)}{x^{2n+2}}}{\biggr )}f(x)\\&={\frac {x^{2}p'_{n}(x)-(2nx-1)p_{n}(x)}{x^{2n+2}}}f(x)\\&={\frac {p_{n+1}(x)}{x^{2(n+1)}}}f(x),\end{aligned}}}</annotation>

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</math><img src="./6f6b08559b09c3b7d787edc00b928a16b5035fc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:48.074ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}f^{(n+1)}(x)&={\biggl (}{\frac {p'_{n}(x)}{x^{2n}}}-2n{\frac {p_{n}(x)}{x^{2n+1}}}+{\frac {p_{n}(x)}{x^{2n+2}}}{\biggr )}f(x)\\&={\frac {x^{2}p'_{n}(x)-(2nx-1)p_{n}(x)}{x^{2n+2}}}f(x)\\&={\frac {p_{n+1}(x)}{x^{2(n+1)}}}f(x),\end{aligned}}}" loading="lazy"></dd></dl>

where pn+1(x) is a polynomial of degree n = (n + 1) − 1. Of course, the (n + 1)st derivative of f is zero for x < 0. For the right-hand side derivative of f (n) at x = 0 we obtain with the above limit

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<annotation encoding="application/x-tex">{\displaystyle \lim _{x\searrow 0}{\frac {f^{(n)}(x)-f^{(n)}(0)}{x-0}}=\lim _{x\searrow 0}{\frac {p_{n}(x)}{x^{2n+1}}}\,e^{-1/x}=0.}</annotation>

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</math><img src="./c468c285f9ed12452954e362480a9a09def3dbdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.944ex; height:6.509ex;" alt="{\displaystyle \lim _{x\searrow 0}{\frac {f^{(n)}(x)-f^{(n)}(0)}{x-0}}=\lim _{x\searrow 0}{\frac {p_{n}(x)}{x^{2n+1}}}\,e^{-1/x}=0.}" loading="lazy"></dd></dl>

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<div class="mw-heading mw-heading3"><h3 id="The_function_is_not_analytic">The function is not analytic</h3></div>

As seen earlier, the function f is smooth, and all its derivatives at the <a href="Origin_(mathematics)" title="Origin (mathematics)">origin</a> are 0. Therefore, the <a href="Taylor_series" title="Taylor series">Taylor series</a> of f at the origin converges everywhere to the <a href="Zero_function" class="mw-redirect" title="Zero function">zero function</a>,

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\frac {f^{(n)}(0)}{n!}}x^{n}=\sum _{n=0}^{\infty }{\frac {0}{n!}}x^{n}=0,\qquad x\in \mathbb {R} ,}</annotation>

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</math><img src="./d49c59be647cf23f64fffa6cd23535cd14715817.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.618ex; height:7.176ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\frac {f^{(n)}(0)}{n!}}x^{n}=\sum _{n=0}^{\infty }{\frac {0}{n!}}x^{n}=0,\qquad x\in \mathbb {R} ,}" loading="lazy"></dd></dl>

and so the Taylor series does not equal f(x) for x > 0. Consequently, f is not <a href="Analytic_function" title="Analytic function">analytic</a> at the origin.

<div class="mw-heading mw-heading3"><h3 id="Smooth_transition_functions">Smooth transition functions</h3></div>

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

<div class="mw-heading mw-heading2"><h2 id="A_smooth_function_that_is_nowhere_real_analytic">A smooth function that is nowhere real analytic</h2></div>

A more <a href="Pathological_(mathematics)" title="Pathological (mathematics)">pathological</a> example is an infinitely differentiable function which is not analytic at any point. It can be constructed by means of a <a href="Fourier_series" title="Fourier series">Fourier series</a> as follows. Define for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a9c6d458566aec47a7259762034790c8981aefab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {R} }" loading="lazy">

</mrow>

</munder>

<msup>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x):=\sum _{k\in \mathbb {N} }e^{-{\sqrt {2^{k}}}}\cos(2^{k}x)\ .}</annotation>

</semantics>

</math><img src="./58607241baf3c85d97a292c9c13832ed1a45c6a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.402ex; height:6.176ex;" alt="{\displaystyle F(x):=\sum _{k\in \mathbb {N} }e^{-{\sqrt {2^{k}}}}\cos(2^{k}x)\ .}" loading="lazy"></dd></dl>

Since the series <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k\in \mathbb {N} }e^{-{\sqrt {2^{k}}}}{(2^{k})}^{n}}">

</mrow>

</munder>

<msup>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</msup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{k\in \mathbb {N} }e^{-{\sqrt {2^{k}}}}{(2^{k})}^{n}}</annotation>

</semantics>

</math><img src="./bc2fd6c7af7e0077fb0b306f6e1fde4b74618c5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.94ex; height:6.176ex;" alt="{\displaystyle \sum _{k\in \mathbb {N} }e^{-{\sqrt {2^{k}}}}{(2^{k})}^{n}}" loading="lazy"> converges for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy">, this function is easily seen to be of class C∞, by a standard inductive application of the <a href="Weierstrass_M-test" title="Weierstrass M-test">Weierstrass M-test</a> to demonstrate <a href="Uniform_convergence" title="Uniform convergence">uniform convergence</a> of each series of derivatives.

We now show that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./71a82805d469cdfa7856c11d6ee756acd1dc7174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}" loading="lazy"> is not analytic at any <a href="Dyadic_rational" title="Dyadic rational">dyadic rational</a> multiple of π, that is, at any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x:=\pi \cdot p\cdot 2^{-q}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x:=\pi \cdot p\cdot 2^{-q}}</annotation>

</semantics>

</math><img src="./dbb8a4d8194b54a6430c4ddccedf65e7770e4b00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.364ex; height:2.843ex;" alt="{\displaystyle x:=\pi \cdot p\cdot 2^{-q}}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d29b9c8775da833d554991bfefdfc54a53085728.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.65ex; height:2.509ex;" alt="{\displaystyle p\in \mathbb {Z} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b67f03c27aecd54c188bc7c10153a0d56bdd516f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.588ex; height:2.509ex;" alt="{\displaystyle q\in \mathbb {N} }" loading="lazy">. Since the sum of the first <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> terms is analytic, we need only consider <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{>q}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./73dbdaa6921d9069f12e3370cd1af8c7076e1167.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.901ex; height:3.009ex;" alt="{\displaystyle F_{>q}(x)}" loading="lazy">, the sum of the terms with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>q}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9fdf1b63991086e94e8a7b183f9908eee528cdf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.379ex; height:2.509ex;" alt="{\displaystyle k>q}" loading="lazy">. For all orders of derivation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2^{m}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ab14407b66fdd78b64c1eeec8dff77a7df81b101.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.331ex; height:2.343ex;" alt="{\displaystyle n=2^{m}}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./42411e85d874a733209223302bbd8d5e3ad04cb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle m\in \mathbb {N} }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\geq 2}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./eca2e437e89ef4565a87f1a6d90ed37eef1d8ce3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.301ex; height:2.343ex;" alt="{\displaystyle m\geq 2}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m>q/2}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f71c319c0b1c0a137f311f1aafc98d0b536ca079.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.533ex; height:2.843ex;" alt="{\displaystyle m>q/2}" loading="lazy"> we have

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{>q}^{(n)}(x):=\sum _{k\in \mathbb {N} \atop k>q}e^{-{\sqrt {2^{k}}}}{(2^{k})}^{n}\cos(2^{k}x)=\sum _{k\in \mathbb {N} \atop k>q}e^{-{\sqrt {2^{k}}}}{(2^{k})}^{n}\geq e^{-n}n^{2n}\quad (\mathrm {as} \;n\to \infty )}">

</mrow>

</mrow>

</msubsup>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</munder>

<msup>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</msup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</munder>

<msup>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</msup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{>q}^{(n)}(x):=\sum _{k\in \mathbb {N} \atop k>q}e^{-{\sqrt {2^{k}}}}{(2^{k})}^{n}\cos(2^{k}x)=\sum _{k\in \mathbb {N} \atop k>q}e^{-{\sqrt {2^{k}}}}{(2^{k})}^{n}\geq e^{-n}n^{2n}\quad (\mathrm {as} \;n\to \infty )}</annotation>

</semantics>

</math><img src="./c5507e13f317b6cad4b039ba3bc7aa4505f5099d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:77.366ex; height:7.509ex;" alt="{\displaystyle F_{>q}^{(n)}(x):=\sum _{k\in \mathbb {N} \atop k>q}e^{-{\sqrt {2^{k}}}}{(2^{k})}^{n}\cos(2^{k}x)=\sum _{k\in \mathbb {N} \atop k>q}e^{-{\sqrt {2^{k}}}}{(2^{k})}^{n}\geq e^{-n}n^{2n}\quad (\mathrm {as} \;n\to \infty )}" loading="lazy"></dd></dl>

where we used the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(2^{k}x)=1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./136dc52f3588c03715b5538d3b6b6cbc54e058fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.762ex; height:3.176ex;" alt="{\displaystyle \cos(2^{k}x)=1}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}>2^{q}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9bcab8b1eda10b4b70ea1b21e4e51f86279410df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.501ex; height:2.676ex;" alt="{\displaystyle 2^{k}>2^{q}}" loading="lazy">, and we bounded the first sum from below by the term with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}=2^{2m}=n^{2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./243b2007d125425e4192abd65b7d34830144fed0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.557ex; height:2.676ex;" alt="{\displaystyle 2^{k}=2^{2m}=n^{2}}" loading="lazy">. As a consequence, at any such <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</mrow>

</munder>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \limsup _{n\to \infty }\left({\frac {|F_{>q}^{(n)}(x)|}{n!}}\right)^{1/n}=+\infty \,,}</annotation>

</semantics>

</math><img src="./fa223a7ee95b25044b6688ef303c8e66071f527c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:32.281ex; height:8.843ex;" alt="{\displaystyle \limsup _{n\to \infty }\left({\frac {|F_{>q}^{(n)}(x)|}{n!}}\right)^{1/n}=+\infty \,,}" loading="lazy"></dd></dl>

so that the <a href="Radius_of_convergence" title="Radius of convergence">radius of convergence</a> of the <a href="Taylor_series" title="Taylor series">Taylor series</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{>q}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./43f4f7c02ea87dde7617f446874f5dd5bbc1c5c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.762ex; height:2.843ex;" alt="{\displaystyle F_{>q}}" loading="lazy"> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is 0 by the <a href="Cauchy-Hadamard_theorem" class="mw-redirect" title="Cauchy-Hadamard theorem">Cauchy-Hadamard formula</a>. Since the set of analyticity of a function is an open set, and since dyadic rationals are <a href="Dense_set" title="Dense set">dense</a>, we conclude that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{>q}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./43f4f7c02ea87dde7617f446874f5dd5bbc1c5c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.762ex; height:2.843ex;" alt="{\displaystyle F_{>q}}" loading="lazy">, and hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

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

</semantics>

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

</mrow>

</mstyle>

</mrow>

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

</semantics>

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

<div class="mw-heading mw-heading2"><h2 id="Application_to_Taylor_series">Application to Taylor series</h2></div>

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

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

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

</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Borel's_lemma" title="Borel's lemma">Borel's lemma</a></div>

For every sequence α0, α1, α2, . . . of real or <a href="Complex_number" title="Complex number">complex numbers</a>, the following construction shows the existence of a smooth function F on the real line which has these numbers as derivatives at the origin.<a href="#cite_note-1">[1]</a> In particular, every sequence of numbers can appear as the coefficients of the <a href="Taylor_series" title="Taylor series">Taylor series</a> of a smooth function. This result is known as <a href="Borel's_lemma" title="Borel's lemma">Borel's lemma</a>, after <a href="%C3%89mile_Borel" title="Émile Borel">Émile Borel</a>.

With the smooth transition function g as above, define

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(x)=g(2+x)\,g(2-x),\qquad x\in \mathbb {R} .}</annotation>

</semantics>

</math><img src="./3a1ac0b6ccd00cb121e697ca4b035c372518cdc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.654ex; height:2.843ex;" alt="{\displaystyle h(x)=g(2+x)\,g(2-x),\qquad x\in \mathbb {R} .}" loading="lazy"></dd></dl>

This function h is also smooth; it equals 1 on the closed interval [−1,1] and vanishes outside the open interval (−2,2). Using h, define for every natural number n (including zero) the smooth function

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{n}(x)=x^{n}\,h(x),\qquad x\in \mathbb {R} ,}</annotation>

</semantics>

</math><img src="./01a6bf09386781eddf40c28ebe222d58510aee12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.557ex; height:2.843ex;" alt="{\displaystyle \psi _{n}(x)=x^{n}\,h(x),\qquad x\in \mathbb {R} ,}" loading="lazy"></dd></dl>

which agrees with the <a href="Monomial" title="Monomial">monomial</a> xn on [−1,1] and vanishes outside the interval (−2,2). Hence, the k-th derivative of ψn at the origin satisfies

</mrow>

</mrow>

</msubsup>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext>otherwise,</mtext>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{n}^{(k)}(0)={\begin{cases}n!&{\text{if }}k=n,\\0&{\text{otherwise,}}\end{cases}}\quad k,n\in \mathbb {N} _{0},}</annotation>

</semantics>

</math><img src="./73b3752e0874884a424619861b8359afc36e0a3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.217ex; height:6.176ex;" alt="{\displaystyle \psi _{n}^{(k)}(0)={\begin{cases}n!&{\text{if }}k=n,\\0&{\text{otherwise,}}\end{cases}}\quad k,n\in \mathbb {N} _{0},}" loading="lazy"></dd></dl>

and the <a href="Boundedness_theorem" class="mw-redirect" title="Boundedness theorem">boundedness theorem</a> implies that ψn and every derivative of ψn is bounded. Therefore, the constants

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

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</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda _{n}=\max {\bigl \{}1,|\alpha _{n}|,\|\psi _{n}\|_{\infty },\|\psi _{n}^{(1)}\|_{\infty },\ldots ,\|\psi _{n}^{(n)}\|_{\infty }{\bigr \}},\qquad n\in \mathbb {N} _{0},}</annotation>

</semantics>

</math><img src="./26b5f8ed3961e53538ad24e49754e865c512bd27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:63.022ex; height:3.676ex;" alt="{\displaystyle \lambda _{n}=\max {\bigl \{}1,|\alpha _{n}|,\|\psi _{n}\|_{\infty },\|\psi _{n}^{(1)}\|_{\infty },\ldots ,\|\psi _{n}^{(n)}\|_{\infty }{\bigr \}},\qquad n\in \mathbb {N} _{0},}" loading="lazy"></dd></dl>

involving the <a href="Supremum_norm" class="mw-redirect" title="Supremum norm">supremum norm</a> of ψn and its first n derivatives, are well-defined real numbers. Define the scaled functions

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<annotation encoding="application/x-tex">{\displaystyle f_{n}(x)={\frac {\alpha _{n}}{n!\,\lambda _{n}^{n}}}\psi _{n}(\lambda _{n}x),\qquad n\in \mathbb {N} _{0},\;x\in \mathbb {R} .}</annotation>

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</math><img src="./ae2dc1a96d81d1f5178f6d401f2bc75d56a48897.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.699ex; height:5.176ex;" alt="{\displaystyle f_{n}(x)={\frac {\alpha _{n}}{n!\,\lambda _{n}^{n}}}\psi _{n}(\lambda _{n}x),\qquad n\in \mathbb {N} _{0},\;x\in \mathbb {R} .}" loading="lazy"></dd></dl>

By repeated application of the <a href="Chain_rule" title="Chain rule">chain rule</a>,

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<annotation encoding="application/x-tex">{\displaystyle f_{n}^{(k)}(x)={\frac {\alpha _{n}}{n!\,\lambda _{n}^{n-k}}}\psi _{n}^{(k)}(\lambda _{n}x),\qquad k,n\in \mathbb {N} _{0},\;x\in \mathbb {R} ,}</annotation>

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</math><img src="./e7be83f0d8431da7e6b937156dc772aea5bf909d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.56ex; height:5.509ex;" alt="{\displaystyle f_{n}^{(k)}(x)={\frac {\alpha _{n}}{n!\,\lambda _{n}^{n-k}}}\psi _{n}^{(k)}(\lambda _{n}x),\qquad k,n\in \mathbb {N} _{0},\;x\in \mathbb {R} ,}" loading="lazy"></dd></dl>

and, using the previous result for the k-th derivative of ψn at zero,

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<mtext>otherwise,</mtext>

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<annotation encoding="application/x-tex">{\displaystyle f_{n}^{(k)}(0)={\begin{cases}\alpha _{n}&{\text{if }}k=n,\\0&{\text{otherwise,}}\end{cases}}\qquad k,n\in \mathbb {N} _{0}.}</annotation>

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</math><img src="./c5d7085231b546b16aee6371d713eb2428ba6f28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.011ex; height:6.176ex;" alt="{\displaystyle f_{n}^{(k)}(0)={\begin{cases}\alpha _{n}&{\text{if }}k=n,\\0&{\text{otherwise,}}\end{cases}}\qquad k,n\in \mathbb {N} _{0}.}" loading="lazy"></dd></dl>

It remains to show that the function

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<annotation encoding="application/x-tex">{\displaystyle F(x)=\sum _{n=0}^{\infty }f_{n}(x),\qquad x\in \mathbb {R} ,}</annotation>

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</math><img src="./2b8913aa1d4580fbdf524841ea12eac2ee785409.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.391ex; height:6.843ex;" alt="{\displaystyle F(x)=\sum _{n=0}^{\infty }f_{n}(x),\qquad x\in \mathbb {R} ,}" loading="lazy"></dd></dl>

is well defined and can be differentiated term-by-term infinitely many times.<a href="#cite_note-2">[2]</a> To this end, observe that for every k

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }\|f_{n}^{(k)}\|_{\infty }\leq \sum _{n=0}^{k+1}{\frac {|\alpha _{n}|}{n!\,\lambda _{n}^{n-k}}}\|\psi _{n}^{(k)}\|_{\infty }+\sum _{n=k+2}^{\infty }{\frac {1}{n!}}\underbrace {\frac {1}{\lambda _{n}^{n-k-2}}} _{\leq \,1}\underbrace {\frac {|\alpha _{n}|}{\lambda _{n}}} _{\leq \,1}\underbrace {\frac {\|\psi _{n}^{(k)}\|_{\infty }}{\lambda _{n}}} _{\leq \,1}<\infty ,}</annotation>

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</math><img src="./f5dcb12db3e03bf7016bfd0212219535288e7deb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:73.322ex; height:10.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }\|f_{n}^{(k)}\|_{\infty }\leq \sum _{n=0}^{k+1}{\frac {|\alpha _{n}|}{n!\,\lambda _{n}^{n-k}}}\|\psi _{n}^{(k)}\|_{\infty }+\sum _{n=k+2}^{\infty }{\frac {1}{n!}}\underbrace {\frac {1}{\lambda _{n}^{n-k-2}}} _{\leq \,1}\underbrace {\frac {|\alpha _{n}|}{\lambda _{n}}} _{\leq \,1}\underbrace {\frac {\|\psi _{n}^{(k)}\|_{\infty }}{\lambda _{n}}} _{\leq \,1}<\infty ,}" loading="lazy"></dd></dl>

where the remaining infinite series converges by the <a href="Ratio_test" title="Ratio test">ratio test</a>.

<div class="mw-heading mw-heading2"><h2 id="Application_to_higher_dimensions">Application to higher dimensions</h2></div>

For every radius r > 0,

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}\ni x\mapsto \Psi _{r}(x)=f(r^{2}-\|x\|^{2})}</annotation>

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</math><img src="./a69703354fabf481dacf0768cff5e0c3e40725c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.44ex; height:3.176ex;" alt="{\displaystyle \mathbb {R} ^{n}\ni x\mapsto \Psi _{r}(x)=f(r^{2}-\|x\|^{2})}" loading="lazy"></dd></dl>

with <a href="Euclidean_norm" class="mw-redirect" title="Euclidean norm">Euclidean norm</a> ||x|| defines a smooth function on n-dimensional <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> with <a href="Support_(mathematics)" title="Support (mathematics)">support</a> in the <a href="Ball_(mathematics)" title="Ball (mathematics)">ball</a> of radius r, but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{r}(0)>0}">

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

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</math><img src="./42c11d1296dca30609c25b7edd47e77abe51da05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.015ex; height:2.843ex;" alt="{\displaystyle \Psi _{r}(0)>0}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Complex_analysis">Complex analysis</h2></div>

This pathology cannot occur with differentiable <a href="Complex_analysis" title="Complex analysis">functions of a complex variable</a> rather than of a real variable. Indeed, all <a href="Holomorphic_functions_are_analytic" class="mw-redirect" title="Holomorphic functions are analytic">holomorphic functions are analytic</a>, so that the failure of the function f defined in this article to be analytic in spite of its being infinitely differentiable is an indication of one of the most dramatic differences between real-variable and complex-variable analysis.

Note that although the function f has derivatives of all orders over the real line, the <a href="Analytic_continuation" title="Analytic continuation">analytic continuation</a> of f from the positive half-line x > 0 to the <a href="Complex_plane" title="Complex plane">complex plane</a>, that is, the function

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} \setminus \{0\}\ni z\mapsto e^{-{\frac {1}{z}}}\in \mathbb {C} ,}</annotation>

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</math><img src="./787e3f10a9f0013aaa96002fb36f603a2e9c2879.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.166ex; height:3.843ex;" alt="{\displaystyle \mathbb {C} \setminus \{0\}\ni z\mapsto e^{-{\frac {1}{z}}}\in \mathbb {C} ,}" loading="lazy"></dd></dl>

has an <a href="Essential_singularity" title="Essential singularity">essential singularity</a> at the origin, and hence is not even continuous, much less analytic. By the <a href="Great_Picard_theorem" class="mw-redirect" title="Great Picard theorem">great Picard theorem</a>, it attains every complex value (with the exception of zero) infinitely many times in every neighbourhood of the origin.

<ul><li><a href="Bump_function" title="Bump function">Bump function</a></li>

<li><a href="Fabius_function" title="Fabius function">Fabius function</a></li>

<li><a href="Flat_function" title="Flat function">Flat function</a></li>

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

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

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> Exercise 12 on page 418 in <a href="Walter_Rudin" title="Walter Rudin">Walter Rudin</a>, Real and Complex Analysis. McGraw-Hill, New Delhi 1980, <style data-mw-deduplicate="TemplateStyles:r1238218222">

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

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

</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-099557-5</bdi>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> See e.g. Chapter V, Section 2, Theorem 2.8 and Corollary 2.9 about the differentiability of the limits of sequences of functions in <cite id="CITEREFAmannEscher2005" class="citation cs2">Amann, Herbert; Escher, Joachim (2005), Analysis I, Basel: <a href="Birkh%C3%A4user_Verlag" class="mw-redirect" title="Birkhäuser Verlag">Birkhäuser Verlag</a>, pp. 373–374, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-7643-7153-6</bdi></cite>

</li>

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

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://planetmath.org/InfinitelydifferentiableFunctionThatIsNotAnalytic">"Infinitely-differentiable function that is not analytic"</a>. <a href="PlanetMath" title="PlanetMath">PlanetMath</a>.</cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-12-24" href="https://en.wikipedia.org/wiki/?title=Non-analytic_smooth_function&oldid=1264871065">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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