Exercise 3, § 2

Let $X, Y, Z$ be three metric spaces and let $H$ be a subset of $\mathscr{C}(X \times Y; Z)$. Suppose that, for each $x_0 \in X$, the set of mappings $u(x_0, .)$, as $u$ runs through $H$, is an equicontinuous subset of $\mathscr{C}(Y; Z)$, and that, for each $y_0 \in Y$, the set of mappings $u(., y_0)$, as $u$ runs through

H, is an equicontinuous subset of $\mathscr{C}(X; Z)$. Show that if X is complete, then for each $b \in Y$ there is a set $S_b$ in X whose complement is meagre in X and which is such that, for each $a \in S_b$, the set H is equicontinuous at $(a, b)$. (Apply Chapter IX, § 5, Exercise 23, using Chapter X, § 2, Proposition 1, Corollary 1.)