Exercise 8, § 3

a) Let f be the continuous mapping $(x, y) \to xy$ of $\mathbf{R} \times \mathbf{R}$ into $\mathbf{R}$. Show that the mapping $x \to f(x, .)$ of $\mathbf{R}$ into $\mathscr{C}_u(\mathbf{R}; \mathbf{R})$ is not continuous.

b) Let X, Y be two topological spaces, let Z be a uniform space and let $f : X \times Y \to Z$ be a mapping. Show that if $f(x, .)$ is continuous on Y for all $x \in X$ and if the mapping $x \to f(x, .)$ of X into $\mathscr{C}_u(Y; Z)$ is continuous, then f is continuous on $X \times Y$.