This comes as a complement to: Relation between total variation and absolute continuity; I was wondering if the following holds:
Let $F$ be a function of bounded variation on $[a,b]$, then $\int_{a}^{b}{|F'(x)|dx} = T_{F}(a,b)$ implies $F$ is absolutely continuous (same notations).
Any help is welcomed.
I guess that we actually have that if $G$ is an increasing continuous function for which $G'(x) < \infty$ a.e, then $G$ is absolutely continuous. (?)