# Real valued function whose derivative is nowhere continuous?

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

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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.
To say that a function is continuous on a set $D$ could mean either (1) that its restriction to $D$ is continuous or (2) that the whole function (not restricted) is continuous at each point of $D$ (i.e., for all $x\in D$ and $\epsilon>0$ there is $\delta>0$ such that, for all $y$ within $\delta$ of $x$ etc. --- whether or not $y\in D$). The result about functions of the first Baire class uses (1), but it's plausible that the OP intended (2). – Andreas Blass Feb 12 '12 at 23:14