I am trying to prove the fundamental theorem of calculus for the Lebesgue integral, but with some kind of different assumptions. Instead of taking $f$ as an absolutely continous function in $[a,b]$, I have $f$ as a continous function, differentiable everywhere on [a,b] and with a continous derivative.
I've read somewhere, that being differentiable everywhere in $[a,b]$ would imply absolute continuity, since $[a,b]$ is a closed interval and, from this, the proof would come clear to me. Is it right? In a positive case, why? In a negative way, how could I work it out?
