Exercise 6, § 1

a) Let $X$ be a topological space, $Y$ a uniform space, and let $\Phi$ be a filter on the set $\mathscr{C}(X; Y)$ which converges pointwise to a function $u_0$. Then $u_0$ is continuous at a point $x_0 \in X$ if and only if, for each entourage $V$ of $Y$ and each set $M \in \Phi$, there is a neighbourhood $U$ of $x_0$ and a mapping $u \in M$ such that $(u_0(x), u(x)) \in V$ for all $x \in U$.

b) Let $X$ be a quasi-compact space, $Y$ a uniform space and $\Phi$ a filter on $\mathscr{C}(X; Y)$ which converges pointwise to a function $u_0$. Then $u_0$ is continuous on $X$ if and only if, for each entourage $V$ of $Y$ and each set $M \in \Phi$, there exists a finite number of functions $u_i \in M (1 \leq i \leq n)$ such that, for each $x \in X$, we have $(u_0(x), u_i(x)) \in V$ for at least one index $i$.