Let $f:[a, b] \rightarrow \mathbb{R}$ be a function that is differentiable everywhere. What can we say about the continuity of $f^{(1)}$? The only results that is related to this I that I can find is that if $f^{(1)}(a) < \lambda < f^{(1)}(b)$, then there is $x\in (a,b)$ s.t. $f(x) = \lambda$ and so $f^{(1)}$ may not have any discontinuities of the first kind.
An example of a function that has a derivative that is discontinuous is $f(x) = x^{2}\sin(\frac{1}{x})$ $ \text{for } x\neq{0}$ and $f(x)=0$ if $x=0$ which has a derivative that has a discontinuity of the second kind at $0$.
I'm not sure how to use this example to create other examples, say $f^{(1)}$ is discontinuous on a: dense subset of $[a,b]$, on a subset of measure $\frac{b-a}{2}$, etc.
Also what happens if we replace " differentiable everywhere" by "differentiable a.e."?
