I am teaching a course on locally compact groups and their representation theory and I am at a point where I would like to introduce continuous representations as generally as possible and provide meaningful examples. The end goal is of course to introduce the left [right] regular representation of $G$ on $L^2(G) = L^2(G, \mu; \mathbb{C})$, where $G$ is locally compact and $\mu$ is Haar measure, but I was hoping to be able to start with other (non-Hilbert) function spaces which might be more natural from a purely topological perspective—for instance, the space $C(G) = C(G; \mathbb{C})$ of continuous complex-valued functions on $G$, equipped with the compact-open topology, and where the action of $G$ is by left (or right) translation. However, I'm not nearly familiar enough with (or sufficiently well-versed in) the compact-open topology to prove that this action is actually continuous—or to discern whether it is only continuous for certain (e.g., locally compact) $G$. I thought I had an argument for compact $G$ but I know realize that it was incomplete. A reference would be greatly appreciated.

Even if the action on $C(G)$ should turn out to not be continuous in general, this would still be instructive to know. My next question would then be if the restricted action on $C_c(G)$, equipped with any of the (possibly very different) topologies considered here, is continuous for $G$ locally compact; it should definitely be true for the topology coming from the inclusion $C_c(G) \hookrightarrow L^2(G)$, but I don't know about the other topologies (provided they are indeed different).