Consider a topological group $G$ (or smooth Lie group) and a topological space $M$ (or smooth manifold) and a group homomorphism $\phi:G\rightarrow Sym(M)$, where $Sym(M)$ is the symmetry group of M, i.e. the group of all homeomorphisms (or diffeomorphisms).
Now consider the map
$$\phi':G\times M\rightarrow M, (g,x)\mapsto \left(\phi(g)\right)(x).$$
I am wondering, if the map $\phi'$ is continuous (or smooth)?
I have recognized that if $G$ is a discrete group, then $\phi'$ is continuous. But I would like to know what happens in the more general case?!
Maybe one needs to impose more assumptions on $G,M$ etc. in order to make the map $\phi'$ continuous (smooth) like the case of discrete topology. But I'm not sure about that.