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What can be said in general of an infinitely smooth function whose Taylor series diverges? According to Borel Theorem it is possible to construct such but what kind of property have those special functions ?

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  • $\begingroup$ See here for an example. $\endgroup$ – Jean-Claude Arbaut Aug 22 '14 at 16:20
  • $\begingroup$ Could you please be more specific about what types of properties you're interested in? The questions is rather vague. $\endgroup$ – Jonas Meyer Aug 22 '14 at 16:23
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Infinitely differentiable but not analytic functions are not "special" at all. In a certain sense, most $C^\infty$ functions are nowhere analytic. See, for example, the presentation About generic properties of "nowhere analyticity" by Esser (joint work with Bastin and Nicolay).

Due to their ubiquity, non-analytic functions do not have any special properties: it's the analytic functions that are special. Of course, one can state the negation of any sufficient condition for analyticity. For example: every extension of a non-analytic function at $x_0$ to a neighborhood of $x_0$ is $\mathbb C$ will fail to be complex-differentiable.

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  • $\begingroup$ Nice slides. Thanks! I've been compiling information on this topic for a bit now, and they list several references and a few results I was not aware of. $\endgroup$ – Andrés E. Caicedo Sep 7 '14 at 6:33

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