Let $f_n$: $(0,1)\to (0,1)$ be given, where $f_n$ is a.e. differentiable, absolutely continuous, and monotone increasing. Suppose $f_n\to f$ uniformly. Then would it implies that $f_n'\to f'$ a.e.?
I know generally uniformly convergence has no way to imply convergence of derivative, but I somehow remembered that monotone increasing would help, but I can't recall where I saw this statement.
PS: I also have $f_n'$ are uniformly bounded (but may not continuous). So I think my statement should hold, right?