Exercise 9, § 3

a) Let X, Y, Z be three topological spaces. Suppose that X and Y are Hausdorff and that every point of each of these spaces has a countable fundamental system of neighbourhoods. Let f be a mapping of $X \times Y$ into Z such that $f(x, .)$ is continuous on Y for each $x \in X$ and such that the mapping $x \to f(x, .)$ of X into $\mathscr{C}_c(Y; Z)$ is continuous. Show that f is continuous on $X \times Y$ (cf. § 1, no. 6, Theorem 2, Corollary 3).

b) Show that the conclusion of a) holds good when the hypothesis on X is replaced by the hypothesis that X is locally compact and Z is a uniform space (use § 2, Exercise 2, and Ascoli's theorem) [cf. Exercise 11 b)].