TITLE: Real valued function whose derivative is nowhere continuous?
QUESTION [6 upvotes]: Let $f:\mathbb{R} \rightarrow \mathbb{R}$. Is it possible that $f$ has a derivative that nowhere continuous on its domain? Please provide an example if possible.
Thanks

REPLY [11 votes]: I would say "no".  $f$ is continuous, and that derivative is therefore of first Baire class.  Such a function is continuous except on a set of first category.  
Now I'll investigate Pietro's reference and see where (or if) I went wrong.
 (checked)  In fact, Pompeiu's derivative is continuous on a dense $G_\delta$
set (where it vanishes) after all.