real analysis - Differentiable function with nowhere differentiable derivitive

I was wondering if it there is a function $f:\mathbb{R} \rightarrow \mathbb{R}$ that is differentiable everywhere, but with a derivitive that is nowhere differentiable.

Answer

Look up your favorite example of a continuous-but-nowhere differentiable function, then integrate it. By the fundamental theorem of calculus you'll get the original function back when you differentiate.

http://en.wikipedia.org/wiki/Weierstrass_function

http://www.proofwiki.org/wiki/Continuous_Nowhere_Differentiable_Function