Proposition 4

Let $(A_i)_{i \in I}$ be a family of subsets of a topological space $X$ whose interiors cover $X$, or which is a locally finite closed covering of $X$. Let $f$ be a mapping of $X$ into a topological space $X'$. If the restriction of $f$ to each of the subspaces $A_i$ is continuous, then $f$ is continuous.

For if $F'$ is a closed subset of $X'$ and if $F = \overline{f}(F')$, then $F \cap A_i$ is closed in $A_i$ for each $i \in I$ ($§ 2$, no. 1, Theorem 1) and therefore $F$ is closed in $X$ by Proposition 3 of no. 1; the result now follows from Theorem 1 of $§ 2$, no. 1.