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

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

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

Some comments: An almost everywhere version 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 quite subtle. Note that do not assume that $f’$ is in $L^1$, nor that $f$ is absolutely continuous, so that the fundamental theorem of calculus does not apply.