TITLE: Existence of a smooth function with nowhere converging Taylor series at every point
QUESTION [20 upvotes]: By Borel's theorem, for any sequence of real numbers $a_n,$ there is a $C^{\infty}$-function
$f:\mathbb{R}\to\mathbb{R}$ whose Taylor series at 0 is $\sum a_nx^n.$ In particular, there are $C^{\infty}$-functions whose Taylor series at a point has 0 radius of convergence.
Motivated by this I have the following question. Is there a $C^{\infty}$-function
$f:\mathbb{R}\to\mathbb{R}$  whose Taylor series has 0 radius of convergence at every
point in $\mathbb{R}?$ I realize that this might sound like a homework problem, but, well,
it's not.

REPLY [3 votes]: Good.  I thought something like this might work.
One more question before I get off this subject, which might be harder (or easier!): what about the other extreme?  Is there a smooth function on an interval in $\mathbb R$, not analytic on any subinterval, whose Taylor series at every point has positive radius of convergence?  The Fabius function might be an example, but this is questionable since the $n$'th derivative has maximum $2^{\sigma(n)}$, where $\sigma(n)=\frac{n(n+1)}{2}$, which is not quite good enough using a crude estimate for radius of convergence if there are points where many derivatives are close to the maximum.