Exercise 5, § 1

Let $f$ be a continuous vector function defined on an open interval $I \subset \mathbf{R}$ with values in a normed space E over $\mathbf{R}$, and admitting a right derivative at every point of I.

a) Show that the set of points $x \in I$ such that $f'_d$ is bounded on a neighbourhood of x is an open set dense in I (use th. 2 of Gen. Top., IX, p. 194).

b) Show that the set of points of I where $f'_d$ is continuous is the complement of a meagre subset of I (cf. Gen. Top., IX, p. 255, exerc. 21).