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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. (?)

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    NVM, got it :).2012-04-04

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