Exercise 15, § 10

Let $X$ be a Hausdorff space, $A$ a closed subset of $X$, $f$ a proper mapping of $X - A$ into a locally compact space $Y$, $Y'$ the one-point compactification of $Y$, and $\omega$ the point at infinity in $Y'$. Let $g$ be the mapping of $X$ into $Y'$ which coincides with $f$ on $X - A$ and maps each point of $A$ to $\omega$. Show that $g$ is continuous.