Exercise 14, § 6

Let $f$ be a numerical function defined on a compact interval $I = [a, b]$ of $\mathbf{R}$, integrable on $I$ for Lebesgue measure. If one sets $F(x) = \int_a^x f(t)\,dt$, show that $F$ admits almost everywhere in $I$ a derivative equal to $f$ (reduce to the case that $f \geq 0$; then, using Exer. 13, consider successively the case that $f$ is lower semi-continuous then the general case (cf. Ch. IV, §4, No. 4, Cor. of Th. 3)).