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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
[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.
[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.