Exercise 7, § 6

Let $X$ be a locally compact space and $f$ a lower semi-continuous function on $X$.

a) If $\limsup_{x \to a} f(x) = +\infty$ for every $a \in X$, show that the set $\overline{f}^{-1}(+\infty)$ is dense in $X$ [for each $a \in X$ and each neighbourhood $V$ of $a$, define a sequence $(U_n)$ of relatively compact open sets such that $\overline{U}_{n+1} \subset U_n$ and $f(x) > n$ for all $x \in U_n$].

b) Let $n \to r_n$ be a bijection of $\mathbf{N}$ onto the set of rational numbers belonging to $[0, 1]$, and let $\varphi(x) = 1/\sqrt{|x|}$ [with $\varphi(0) = +\infty$]; then the function $f(x) = \sum_{n=0}^{\infty} 2^{-n} \varphi(x - r_n)$ is lower semi-continuous on $[0, 1]$, $\overline{f}^{-1}(+\infty)$ is dense in this interval, and so is its complement [to prove the last point, observe that the series is convergent whose general term is

(Integration, Chapter IV, § 3, no. 6, Theorem 5).]

c) If $f$ is finite, show that the set of points $x$ such that $\omega(x; f)$ is finite is dense [use a)].

d) Show that the set of points of $X$ at which $f$ is continuous ($f$ being finite or not) is dense. [Reduce to the case in which $f$ is bounded, by replacing $f$ by $f/(1 + |f|)$ (Exercise 4); observe that $1/\omega(x; f)$ is lower semi-continuous and use a) and Exercise 6 c)].