An infinitely differentiable function that is not analytic

The function

Consider the real function

How it is ill-behaved

One can show that f has derivatives of all orders at every point including 0. To show this at x = 0, use L'Hopital's rule, mathematical induction, and some simple substitutions.

[Detail could be put here.]

But in proving this, one will find that f⁽ⁿ⁾(0) = 0 for all n. Therefore, the Taylor series of f is

unless x = 0. Consequently f is not analytic at 0. This pathology cannot occur with functions of a complex variable rather than of a real variable.

How this is negatively a good thing

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

How this is positively a good thing

Via a sequence of piecewise definitions [Details could be put here.] one may construct from this function a function g(x) such that

and further, such that g has derivatives of all orders at every point. By multiplying this by any infinitely differentiable function one can get an infinitely differentiable function with prescribed behavior on the interval [a, b] whose support is bounded. Only by showing the existence of functions with this sort of behavior can one be sure that Schwartz's theory of distributions (or "generalized functions") does not become vacuous for lack of test functions.

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