Differentiable functions such that the derivative is nowhere continuous.

Question

Is there a function $f:\mathbb R\to\mathbb R$ which is differentiable but such that the derivative is nowhere continuous?

Answer 1

It is impossible, because of Baire's simple limit theorem:

If a sequence of continuous functions $f_n\colon\mathbf R\to\mathbf R$ converges pointwise to a function $f$, $f$ is continuous on a dense subset of $\mathbf R$.

As $f'(x)$ is the limit of the sequence $g_n(x)=n\biggl(f\Bigl(x+\dfrac1n\Bigr)-f(x)\biggr)$, Baire's theorem can be applied.