Exercise 7, § 5

Let $f$ be a real-valued function defined on a closed subset $A$ of $\overline{\mathbf{R}}$. Let $\lim \sup_{x \to a, x \geqslant a} f(x)$ [resp. $\lim \sup_{x \to a, x \leqslant a} f(x)$] denote the upper limit of $f(x)$ as $x$ tends to a point $a \in A$ while remaining in $A \cap [a, +\infty]$ (resp. $A \cap [-\infty, a]$). Show that the set of points $a \in A$ such that $\limsup_{x \to a, x \geq a} f(x) \neq \limsup_{x \to a, x \leq a} f(x)$ is countable. [Prove that, for each pair of rational numbers $p, q$ such that $p < q$, the set $G_{p,q}$ of points $a \in A$ such that

is countable, by using Exercise 1 of § 2].