Let $f_n$ be a sequence of differentiable functions on $[0, 1]$ with 

 1. $f_n \to f$ uniformly for some (necessarily) continuous $f$.
 2. $f'_n - g \to 0$ in $L^{\infty}$ for some measurable $g$.
 

Is it true that $f$ is differentiable everywhere, with $f' = g$ almost everywhere?

**Some comments:** An [almost everywhere version](https://mathoverflow.net/questions/471822/is-the-w1-infty-limit-of-differentiable-a-e-functions-also-differentiabl) of the same question is answered negatively here. I expected to have an easy affirmative answer if $f$ are assumed instead everywhere differentiable, but to my surprise this seems to be much more subtle. Note that we do not assume that $f’_n$ are in $L^1$, nor that $f_n$ are absolutely continuous, so that the fundamental theorem of calculus does not apply.