Does there exist a monotone function $f: [0, 1] \to \mathbb R$ that is differentiable everywhere, but its derivative is discontinuous a.e.?



Monotone differentiable function whose derivative is discontinuous on a full measure set
I think you want the [Pompeiu function](https://en.wikipedia.org/wiki/Pompeiu_derivative) (Feel free to add questions on the details of the construction).


