Exercise 4, § 2

Define on the interval [0, 1] two real functions $f, g$ such that $f$ and $g$ admit strict primitives, but that $fg$ is at every point of [0, 1[ the right derivative of a continuous function which has no left derivative at a set of points having the power of the continuum (use constructions similar to those of exerc. 8 of I, p. 38 and exerc. 3 above).