Corollary 2

Let $X$ be a topological space, $\mathscr{S}$ a set of subsets of $X$, and $Y$ a uniform space. Let $\tilde{C}_\mathscr{S}(X; Y)$ denote the set of all mappings of $X$ into $Y$ whose restriction to each set of $\mathscr{S}$ is continuous. Then $\tilde{C}_\mathscr{S}(X; Y)$ is a closed subspace of the uniform space $F_\mathscr{S}(X; Y)$ and is complete if $Y$ is complete.

Suppose that $u$ lies in the closure of $\tilde{C}_\mathscr{S}(X; Y)$ in $F_\mathscr{S}(X; Y)$; then (no. 2), for each $A \in \mathscr{S}$, $u|A$ lies in the closure of $C(A; Y)$ in $F_u(A; Y)$, and is therefore continuous by Theorem 2.