What can be said about an infinitely differentiable function whose Taylor series diverges?

Question

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 ?

Answer 1

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.