Exercise 23, § 5

Let $X$ be a topological space, and let $Y, Z$ be two metric spaces, with metrics $d, d'$ respectively. Let $f : X \times Y \to Z$ be a mapping such that $y \to f(x_0, y)$ is continuous on $Y$ for each $x_0 \in X$ and such that $x \to f(x, y_0)$ is continuous on $X$ for each $y_0 \in Y$.

a) For each real number $\varepsilon > 0$, each $b \in Y$ and each $x \in X$, let $g(x; b, \varepsilon)$ denote the least upper bound of the numbers $\alpha > 0$ such that the relation $d(b,\ y) < \alpha$ implies $d'(f(x,\ b),\ f(x,\ y)) \leq \varepsilon$. Show that the function $x \to g(x;\ b,\ \varepsilon)$ is upper semi-continuous.

b) If $X$ is a Baire space, deduce from a) and Theorem 2 that for each $b \in Y$ there is a set $S_b$ in $X$, whose complement is meagre, such that $f$ is continuous at $(a,\ b)$ for each $a \in S_b$.