Suppose $f: [a,b] \rightarrow \mathbb{C}$ is absolutely continuous and $\exists$ function $g$ continuous, $f'=g$ almost everywhere.
I'm trying to prove that $f$ is differentiable everywhere and $f'(x)=g(x)$ for every $x$.
My thought is to use fundamental theorem of calculus for continuous function $g(x)$ and the fact that f is differentiable a.e and $f' \in \mathcal{L}^1[a,b]$, which implies $\int f'=\int g$. I'm quite confused about what to do next because we haven't really studied the fundamental theorem of calculus in Lebesgue measure point of view. Just wondering if my idea is doable.