Exercise 8, § 6

Let $X$ be a topological space, and let $A$ be a closed subset of $X \times \overline{\mathbf{R}}$. Show that the mapping $x \to \inf (A(x))$ of $\mathrm{pr}_1 A$ into $\overline{\mathbf{R}}$ is lower semi-continuous. Conversely, if $f : X \to \overline{\mathbf{R}}$ is a lower semi-continuous function, then the subset $B$ of $X \times \overline{\mathbf{R}}$ consisting of pairs $(x, y)$ such that $y \geqslant f(x)$ is closed in $X \times \overline{\mathbf{R}}$.