Let $G$ be a group of homeomorphisms of a topological space $X$. The action of $G$ on $X$ is said to be discontinuous at a point $x \in X$ if
$G_x :=$ the stabilizer of $x$, is finite.
$x$ has an open neighbourhood $U$ such that $gU \cap U = \emptyset$ for all $g \notin G_x$
If $G$ is a topological group action on a topological space $X$ then the action is said to be continuous if the map $F : G \times X \rightarrow X$ given by $(g,x) \mapsto gx$ is continuous.
My question is - are these two notions the opposite of each other (considering the way they are named)? That is if $G$ is a topological group acting on $X$ then will the map $F$ not being continuous at some point $(x,g)\in G$ mean that the action is discontinuous at $x \in X$? I can't see why this should be true.
Can some one give me an example of a continuous group action that is discontinuous (if possible)?