Exercise 8, § 1

Let $f$ be a mapping from $\mathbf{R}$ to itself such that the set C of points of $\mathbf{R}$ where $f$ is continuous is dense in $\mathbf{R}$, and such that the complement A of C is also dense. Show that the set D of points of C where $f$ is right differentiable is meagre. (For each integer n, let $E_n$ be the set of points $a \in \mathbf{R}$ such that there exist two points $x, y$ such that $0 < x - a < 1/n, 0 < y - a < 1/n$ and

Show that the interior of $E_n$ is dense in $\mathbf{R}$. For this, note that for every open nonempty interval $I$ in $\mathbf{R}$ there is a point $b \in I \cap A$; show that for $a < b$ and $b - a$ sufficiently small one has $a \in E_n$.)