I've tried but I cannot tell whether the following is true or not. Let $f:[0,1]\rightarrow \mathbb{R}$ be a nondecreasing and continuous function. Is it true that I can find a Lebesgue integrable function $h$ such that $ f(x)=f(0)+\int_{0}^{x}h(x)dx $ such that $f'=h$ almost everywhere?
Any hint on how to proceed is really appreciated it!