Editing Non-analytic smooth function (section)

== Application to Taylor series ==
<!--The article on Taylor series links to this section-->
{{main|Borel's lemma}}
For every sequence α<sub>0</sub>, α<sub>1</sub>, α<sub>2</sub>, .&nbsp;.&nbsp;. of real or [[complex number]]s, the following construction shows the existence of a smooth function ''F'' on the real line which has these numbers as derivatives at the origin.<ref>Exercise 12 on page 418 in [[Walter Rudin]], ''Real and Complex Analysis''. McGraw-Hill, New Delhi 1980, {{isbn|0-07-099557-5}}</ref> In particular, every sequence of numbers can appear as the coefficients of the [[Taylor series]] of a smooth function.  This result is known as [[Borel's lemma]], after [[Émile Borel]].

With the smooth transition function ''g'' as above, define

:<math>h(x)=g(2+x)\,g(2-x),\qquad x\in\mathbb{R}.</math>

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

:<math>\psi_n(x)=x^n\,h(x),\qquad x\in\mathbb{R},</math>

which agrees with the [[monomial]] ''x<sup>n</sup>'' on <nowiki>[</nowiki>&minus;1,1<nowiki>]</nowiki> and vanishes outside the interval (&minus;2,2). Hence, the ''k''-th derivative of ''ψ<sub>n</sub>'' at the origin satisfies

:<math>\psi_n^{(k)}(0)=\begin{cases}n!&\text{if }k=n,\\0&\text{otherwise,}\end{cases}\quad k,n\in\mathbb{N}_0,</math>

and the [[boundedness theorem]] implies that ''ψ<sub>n</sub>'' and every derivative of ''ψ<sub>n</sub>'' is bounded. Therefore, the constants

:<math>\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,</math>

involving the [[supremum norm]] of ''ψ<sub>n</sub>'' and its first ''n'' derivatives, are well-defined real numbers. Define the scaled functions

:<math>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}.</math>

By repeated application of the [[chain rule]],

:<math>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},</math>

and, using the previous result for the ''k''-th derivative of ''ψ<sub>n</sub>'' at zero,

:<math>f_n^{(k)}(0)=\begin{cases}\alpha_n&\text{if }k=n,\\0&\text{otherwise,}\end{cases}\qquad k,n\in\mathbb{N}_0.</math>

It remains to show that the function

:<math>F(x)=\sum_{n=0}^\infty f_n(x),\qquad x\in\mathbb{R},</math>

is well defined and can be differentiated term-by-term infinitely many times.<ref>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 {{Citation
  | last = Amann
  | first = Herbert
  | last2 = Escher
  | first2 = Joachim
  | title = Analysis I
  | place = Basel
  | publisher = [[Birkhäuser Verlag]]
  | year = 2005
  | pages = 373–374
  | isbn = 3-7643-7153-6}}</ref> To this end, observe that for every ''k''

:<math>\sum_{n=0}^\infty\|f_n^{(k)}\|_\infty
\le \sum_{n=0}^{k+1}\frac{|\alpha_n|}{n!\,\lambda_n^{n-k}}\|\psi_n^{(k)}\|_\infty
+\sum_{n=k+2}^\infty\frac1{n!}
\underbrace{\frac1{\lambda_n^{n-k-2}}}_{\le\,1}
\underbrace{\frac{|\alpha_n|}{\lambda_n}}_{\le\,1}
\underbrace{\frac{\|\psi_n^{(k)}\|_\infty}{\lambda_n}}_{\le\,1}
<\infty,</math>

where the remaining infinite series converges by the [[ratio test]].