Let $f : [0,1] \rightarrow [0, + \infty)$. If there exists $g \in L^1([0,1]) $ s.t. for every $t,s \in [0,1]$ holds $$ |f(t) - f(s)| \leq \int_s^t g(u) \, du \quad (t>s),$$ then there exists $h \in L^1([0,1])$ s.t. for almost every $t,s \in [0,1]$ holds $$f(t) - f(s) = \int_s^t h(u) \, du \quad (t>s).$$
Is this true? I suspect that there is a quite straightforward way to show this, but can't figure it out. I suppose that one could prove the statement using absoulte continuity and Lebesgue integral theorem, but I'm actually looking for a direct proof of the statement.