Exercise 16, § 6

a) Let $g$ be a function integrable for Lebesgue measure on $E = [0, 1]$. Set $G(x) = \int_0^x g(t)\,dt$; show that at every point $x \in E$ where $g$ is continuous, $G$ admits a derivative equal to $g$.

b) Give an example of a function $g$ that is bounded and is continuous almost everywhere in $E$, such that $G$ has no right derivative at the points of an uncountable subset of $E$ (cf. FRV, I, §2, Exer. 8).

c) Show that the function equal to $x^2 \sin \frac{1}{x^2}$ for $x \neq 0$, and to 0 for $x = 0$, admits a derivative at every point of $E$, but this derivative is not integrable for Lebesgue measure on $E$.