Let $X$ and $Y$ be $G$-spaces for a group G (you may assume Hausdorff). Let $Y^X$ be the space of continuous mappings from $X$ to $Y$ with the compact open topology. This space carries a conjugation action of $G$.
It is often stated that under some minor regularity assumptions (X locally compact ?), this becomes a G-space, i.e. the action becomes continuous, but I have never seen a proof of this basic fact anywhere. I'd appreciate any hints!