$f$ continuous identical to absolutely continuous $g$ a.e. $\Rightarrow f=g$?

Question

Let $f:[a,b]\to\mathbb{C}$ be a function identical to an absolutely continuous function $g$ almost everywhere. I was wondering whether if $f$ is continuous we can infer that $f$ is absolutely continuous and identical to $g$, but I can find no information on this issue and I have not been able to prove it by myself. Does anybody know more about that?

Answer 1

The set where $f\neq g$ is open if $f$ and $g$ are continous, hence has positive measure or is empty. So yes, if $f=g$ a.e. for continuous $f,g$, then $f=g$. (Assuming you are using Lebesgue measure).

Answer 2

Because $g$ is absolutely continuous, it is continuous; so, because $f$ and $g$ are continuous and $f=g$ a.e., you can prove pretty directly that it must be the case that $f=g$. But, if $f=g$ and $g$ is absolutely continuous, then certainly $f$ is absolutely continuous as well.