When is the image of a proper map closed?

First of all the definition of a proper map assumes continuity by convention (I have not come across texts that say otherwise)

Secondly, here is a more general result -

Lemma : Let $f:X\rightarrow Y$ be a proper map between topological spaces $X$ and $Y$ and let $Y$ be locally compact and Hausdorff. Then $f$ is a closed map.

Proof : Let $C$ be a closed subset of $X$. We need to show that $f(C)$ is closed in $Y$ , or equivalently that $Y\setminus f(C)$ is open.

Let $y\in Y\setminus f(C)$. Then $y$ has an open neighbourhood $V$ with compact closure. Then $f^{-1}(\bar{V})$ is compact.

Let $E=C\cap f^{-1}(\bar{V})$ . Then clearly $E$ is compact and hence so is $f(E)$. Since $Y$ is Hausdorff $f(E)$ is closed.

Let $U=V\setminus f(E)$. Then $U$ is an open neighbourhood of $y$ and is disjoint from $f(C)$.

Thus $Y\setminus f(C)$ is open. $\square$

I hope this helps.

EDIT: To clarify the statement $U$ is disjoint from $f(C)$ -

Suppose $z\in U\cap f(C)$ Then there exists a $c\in C$ such that $z=f(c)$. This means $c\in f^{-1}(U)\subseteq f^{-1}(V)\subseteq f^{-1}(\bar V)$. So $c\in C\cap f^{-1}(\bar V)=E$. So $z=f(c)\in f(E)$ which is a contradiction as $z\in U$.

One may generalize the result in R_D's answer even further:

A proper map $f:X\to Y$ to a compactly generated Hausdorff space is a closed map (A space $Y$ is called compactly generated if any subset $A$ of $Y$ is closed when $A\cap K$ is closed in $K$ for each compact $K\subseteq Y$).
Proof: Let $C\subseteq X$ be closed, and let $K$ be a compact subspace of $Y$. Then $f^{-1}(K)$ is compact, and so is $f^{-1}(K)\cap C =: B$. Then $f(B)=K\cap f(C)$ is compact, and as $Y$ is Hausdorff, $f(B)$ is closed. Since $Y$ is compactly generated, $f(C)$ is closed in $Y$.

A locally compact space $Y$ is compactly generated: If $A\subset Y$ intersects each compact set in a closed set, and if $y\notin A$, then $A$ intersects the compact neighborhood $K$ of $y$ in a closed set $C$. Now $K\setminus C$ is a neighborhood of $y$ disjoint from $A$, hence $A$ is closed.