Is it possible for a continuous function to have a nowhere-continuous derivative?

Question

This is motivated by a question I saw elsewhere that asks whether there is a real-valued function on an interval that contains no monotone subintervals.

Edit: Note that I am asking for a function whose derivative exists but is not continuous anywhere.

Answer 1

Consider the Weierstrass Function

It is continuous everywhere and only differentiable at a set of points with measure 0. I don't know if that suffices for you, but I think it is quite amazing already.

Answer 2

(Adding back someone else's answer that was deleted for some reason. )

Answer: NO: If $f$ is differentiable everywhere on $\mathbb R$, then $f'$ is continuous somewhere.

Suppose $f$ is differentiable everywhere. Then $f$ is continuous everywhere. The functions $$ g_n(x) = \frac{f\left(x+\frac{1}{n}\right)-f(x)}{\frac{1}{n}} $$ are continuous and converge pointwise everywhere to $f'(x)$. Therefore $f'(x)$ is of Baire class $1$, and therefore has lots of points of continuity.