Exercise 3, § 1

Let $\mathscr{C}(I)$ be the complete space of continuous finite real functions defined on the compact interval $I = [a, b]$ of $\mathbf{R}$, and endow $\mathscr{C}(I)$ with the topology of uniform convergence (\emph{Gen. Top.}, X, p. 277). Let $A$ be the subset of $\mathscr{C}(I)$ formed by the functions $x$ such that for \emph{at least one} point $t \in [a, b[$ (depending on the function $x$) the function $x$ has a \emph{finite} right derivative. Show that $A$ is a \emph{meagre} set in $\mathscr{C}(I)$ (\emph{Gen. Top.}, IX, p. 192), and hence its complement, that is, the set of continuous functions on $I$ not having a finite right derivative at \emph{any point} of $[a, b[$ is a Baire subspace of $\mathscr{C}(I)$ (\emph{Gen. Top.}, IX, p. 192). (Let $A_n$ be the set of functions $x \in \mathscr{C}(I)$ such that for at least one value of $t$ satisfying $a \leq t \leq b - 1/n$ (and depending on $x$) one has $|x(t') - x(t)| \leq n |t' - t|$ for all $t'$ such that $t \leq t' < t + 1/n$. Show that each $A_n$ is a closed \emph{nowhere dense} set in $\mathscr{C}(I)$: remark that in $\mathscr{C}(I)$ each ball contains a function having bounded right derivative on $[a, b[$; on the other hand, for every $\varepsilon > 0$ and every integer $m > 0$ there exists on $I$ a continuous function having at every point of $[a, b[$ a finite right derivative such that, for all $t \in [a, b[$ one has $|y(t)| \leq \varepsilon$ and $|y'_r(t)| \geq m.$)