Exercise 9, § 6

a) Let $X$ and $Y$ be two Hausdorff spaces and let $\pi : X \to Y$ be a proper mapping (Chapter I, § 10, no. 1). Let $g$ be a lower semi-continuous real-valued function on $X$. For each $y \in Y$, let $f(y)$ be the greatest lower bound of $g$ in the set $\overline{\pi}^{-1}(y)$ [thus $f(y) = +\infty$ if $\overline{\pi}^{-1}(y) = \varnothing$]. Show that $f$ is lower semi-continuous on $Y$. [Use the fact that $\pi(X)$ is closed, that $\overline{\pi}^{-1}(y)$ is compact for all $y \in \pi(X)$, and that every neighbourhood of $\overline{\pi}^{-1}(y)$ contains a neighbourhood which is saturated with respect to the equivalence relation $\pi(x) = \pi(x')$.]

b) Let $g$ be the continuous function $|x_1 x_2 - 1|$ defined on $\mathbf{R} \times ]0, +\infty[$, and for each $x_1 \in \mathbf{R}$ let $f(x_1) = \inf_{x_2 > 0} g(x_1, x_2)$. Show that $f$ is not lower semi-continuous.