I'm interested to know if a piecewise-continuous monotone function from $\mathbb{R}$ to $\mathbb{R}$ is also piecewise-continuously differentiable. I mean piecewise in the sense that there is a finite or countably infinite and isolated set $S\subset\mathbb{R}$, and intervals covering $\mathbb{R}\setminus S$ such that on any of these intervals, continuity and continuous-differentiability are verified.
I know that
- a monotone function is almost everywhere differentiable (this topic)
- there are monotone and everywhere differentiable functions that are not continuously differentiable (this other topic)
- almost everywhere differentiable may not imply almost everywhere continuously differentiable, the derivative can actually be nowhere continuous (this one)
It seems to me that this third reference is not exactly a counter-example as from what I understand, it features a sort of devil-staircase function and I guess this one is not piecewise-continuous? (possible accumulating sequence of discontinuities)