Continuous group action

Let $G$ be a topological group and $X$ a topological space. A group action $\alpha:G\times X\to X$ is continuous when $\alpha$ is a continuous map for the product topology on $G\times X$.

Equivalently, the action varies continuously both with the acting element and with the point. Every fixed $g\in G$ then acts by a homeomorphism, with inverse given by $g^{-1}$. Continuity is separate from faithfulness: continuity is topological regularity, while faithfulness says that only the identity acts trivially everywhere.