A function [itex]f:\mathbb{R} \to \mathbb{R}[/itex] is called "smooth" if its k-th derivative exists for all k. A function is called analytic at a if its Taylor series(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \sum_{n\geq 0} \frac{f^{(n)}(a)}{n!} (x-a)^n [/tex]

converges and is equal to f(x) in a small neighborhood around a.

The challenge: Construct or otherwise prove the existence of a function which is smooth, but which is not analytic on as large a set as possible.

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# Challenge 14: Smooth is not enough

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