Exercise 6, § 10

Let $X$ be a space which is the sum of subspaces $X_i (i \in I)$, and let $f$ be a continuous mapping of $X$ into a topological space $Y$. For each $i \in I$, let $f_i : X_i \to Y$ be the restriction of $f$ to $X_i$. Show that $f$ is proper if and only if each of the $f_i$ is proper and the family $(f(X_i))_{i \in I}$ is locally finite [use Theorem 1 d)].