An infinitely differentiable function that is not analytic

From Academic Kids

In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not true, with this article constructing a counterexample.

Contents

The function

Consider the function

<math>f(x)=\left\{\begin{matrix}\exp(-1/x^2) & \mbox{if}\ x\neq 0 \\ \\ 0 & \mbox{if}\ x=0 \end{matrix}\right.<math>

where x is a real number.

How it is ill-behaved

f is smooth on R

It can be shown that <math>f<math> admits continuous derivatives of any order in all points of <math>\mathbb{R}<math>, given by

<math>f^{(n)}(x) = \left\{\begin{matrix} R_n(x)\,f(x) & (x \ne 0) \\ 0 & (x = 0)\end{matrix}\right.,<math>

where <math> R_n <math> is a rational function without pole on <math>\mathbb{R}^*=\mathbb{R}\setminus\{0\}<math>.

The proof, by induction, is mainly based on the fact that for any integer <math>m<math>,

<math>\lim_{x \to 0} x^m e^{-1/x^2} = 0,<math>

which implies that all <math>f^{(n)}<math> are continuous and differentiable in <math>x = 0<math>, because

<math>\lim_{x \to 0} \frac{f^{(n)}(x) - f^{(n)}(0)}{x-0} = \lim_{x \to 0} \frac{R_n(x)}{x}\,e^{-1/x^2} = 0<math>.

f is not analytic

According to what precedes, the Taylor series of <math>f<math> is

<math>\sum_{n=0}^\infty {0\over n!}x^n = 0\neq f(x)<math>

unless <math>x = 0<math>. Consequently <math>f<math> is not analytic at 0. This pathology cannot occur with functions of a complex variable rather than of a real variable. Note that although this function has derivatives of all orders over the real line, the analytic continuation of <math>f<math> to the complex plane has an essential singularity at the origin, and hence is not even continuous, much less analytic. Indeed, all holomorphic functions are analytic, so that the failure of <math>f<math> 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.

How this is a good thing...

...in negative terms

This example teaches us that functions of a real variable are sometimes ill-behaved in ways to which functions of a complex variable are immune.

...in positive terms

One of the most important applications of this function is the construction of so-called mollifiers, which are important in theories of generalized functions, like e.g. Laurent Schwartz's theory of distributions.

The existence of these functions represents one of the main differences between differential geometry and analytic geometry. In terms of sheaf theory, this difference can be stated by saying that the sheaf of differentiable functions on a differentiable manifold is flasque, in contrast with the analytic case.

The function above is generally used to build up partitions of unity on differentiable manifolds.

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