Exercise 4, § 1

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

a) Let U be a nonempty open set in E, and A the subset of I formed by the points x such that $f'_d(x) \in U$. Given a number $\alpha > 0$ let B be the subset of I formed by the points x such that there exists at least one $y \in I$ satisfying the conditions $x - \alpha \leq y < x$ and $(f(x) - f(y))/(x - y) \in U$; show that the set B is open and that $A \cap CB$ is countable (remark that this last set is formed by the left-hand endpoints of intervals contiguous to CB). Deduce that the set of points $x \in A$ such that $f'_g(x) \notin \overline{U}$ is countable.

b) Suppose that E is a normed space; the image $f(I)$ is then a metric space having a countable base, and the same is true for the closed vector subspace F of E generated by $f(I)$, a subspace which contains $f'_d(I)$ and $f'_g(I)$. Deduce from a) that the set of points $x \in I$ such that $f'_d(x) \neq f'_g(x)$ is countable. (If $(U_n)$ is a countable base for the topology of F note that for two distinct points $a, b$ of F there exist two disjoint sets $U_p, U_q$ such that $a \in U_p$ and $b \in U_q$.)

c) Take for E the product $\mathbf{R}^I$ (the space of mappings from I into $\mathbf{R}$, endowed with the topology of simple convergence), and for each $x \in I$ denote by $g(x)$ the map $t \mapsto |x - t|$ of I into $\mathbf{R}$. Show that g is continuous and that, for every $x \in I$, one has $g'_r(x) \neq g'_l(x)$.