Let $G$ be a group. A G-module M is defined as an abelian group on which $G$ acts through the map $ G \times M \to M$ where $ (g, m) \mapsto g \cdot m$ This action satisfies $g \cdot (m + m') = g \cdot m + g \cdot m', \forall m, m' \in M$.
When $G$ is a Galois group, a G-module M is described as an abelian group on which $G$ acts continuously, respecting the Krull topology on $G$ and the discrete topology on $M$.
Are these two definitions compatible? In particular, if $G$ acts continuously on $M$, does it necessarily imply that the action $G \times M \to M, (g, m) \mapsto g \cdot m$ satisfies $g \cdot (m + m') = g \cdot m + g \cdot m' , \forall m, m' \in M$?
The definition of a $G$-module for a profinite group is as follows.
Definition: Let $G$ be a profinite group. An abelian group $M$ is called a continuous (or discrete) $G$-module, if it is a $G$-module in the usual sense, and in addition the action $G\times M\rightarrow M$ is continuous when $M$ is endowed with the discrete topology, and $G\times M$ with the product topology.
So in particular, $g(m+m')=gm+gm'$ already holds.
Not every $G$-module $M$ is continuous. Consider the Galois extension $\Bbb Q(\sqrt{\Bbb N})/\Bbb Q$ with Galois group $G$. Then $$ M=\prod_p\Bbb Q(\sqrt{p}) $$ is a $G$-module, which is not continuous.