Exercise 30, § 4

Let $X$ be a normal space and let $g$ (resp. $f$) be an upper (resp. lower) semi-continuous function on $X$, such that $g \leq f$. Show that there is a continuous real-valued function $h$ on $X$ such that $g \leq h \leq f$. [Reduce to the case where $f$ and $g$ take their values in $[0, 1]$. For each dyadic number $r \in [0, 1]$, let $F(r)$ [resp. $G(r)$] be the set of all $x \in X$ such that $f(x) \leq r$ (resp. $g(x) < r$). Define by induction a family of open sets $U(r)$ (where $r$ runs through the set of dyadic numbers contained in $[0, 1]$) such that when $r < r'$ we have $F(r) \subset U(r')$, $\overline{U(r)} \subset U(r')$ and $\overline{U(r)} \subset G(r')$, by imitating the proof of Theorem 1 of no. 1; then complete the proof as in Theorem 1.]