Consider Lebesgue integrals over the real line. I have the following problem:
Problem: Suppose $F(x)$ is a continuous function in $[a,b]$, and $F'(x)$ exists everywhere and is integrable. Show $F(x)$ is absolutely continuous.
The hint in the book suggested showing that $F'(x)\ge 0$ a.e. implies that $F(x)$ is increasing. I did that, but I do not see how it helps with this problem. How can this hint be used to solve the problem?
(I am aware other proofs exist.)