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.



Existence of a smooth function with nowhere converging Taylor series at every point
S.S. Kim and K.H. Kwon gave an explicit example of a *monotone* smooth but nowhere analytic function ([link](http://www.jstor.org/pss/2589322)), which is an anti-derivative of the function
$$\psi(x)=\sum\limits\_{k=1}^{\infty} \frac{1}{k!}\phi(2^k(x-[x])),$$
where
$$\phi(x) = \begin{cases} \exp{\left(-\frac{1}{x^2}-\frac{1}{(x-1)^2}\right)},\qquad & 0 < x < 1,\ \\\ \\\ 0, & \mbox{otherwise.} \end{cases}$$


In fact, the set of smooth but nowhere analytic functions on $\mathbb R$ is of the second category in $C^{\infty}(\mathbb R)$ (just like the set of all continuous but nowhere differentiable functions is of the second category in $C(\mathbb R)$). See a one page note by R. Darst ["*Most infinitely differentiable functions are nowhere analytic*"](http://books.google.co.uk/books?id=k86bShPCxfUC&pg=PA597&lpg=PA597&dq=%2522Most+infinitely+differentiable+functions+are+nowhere+analytic%2522&source=bl&ots=5LvFhmFVwR&sig=U5IxIiA6VBo_W1oPcuIAPqrrbHQ&hl=en&ei=Ff7ETL3cNcufOpW5xYUM&sa=X&oi=book_result&ct=result&resnum=7&ved=0CDcQ6AEwBg#v=onepage&q=%2522Most%2520infinitely%2520differentiable%2520functions%2520are%2520nowhere%2520analytic%2522&f=false).


**Edit.** Kim and Kwon mention in their paper that the first concrete example of smooth  

but nowhere analytic function dates back to A Pringsheim ([*"Zur Theorie der Taylor'schen Reihe und der analytischen Functionen mit beschränktem Existenzbereich."*](http://www.ams.org/mathscinet-getitem?mr=1510771) *Math. Ann.* 42 (**1893**), no. 2, 153–184.)


