Exercise 21, § 2

Let $G$ be a topological group and let $H, K$ be two subgroups of $G$ such that $G = HK$ and $H \cap K = \{e\}$. Each $x \in G$ can be written uniquely in the form $x = f(x)g(x)$, where $f(x) \in H$ and $g(x) \in K$. The mapping $(y, z) \mapsto yz$ of $H \times K$ onto $G$ is a homeomorphism if and only if one of the mappings $f, g$ is continuous. Show that this condition is always satisfied if one of the subgroups $H, K$ is compact and closed, and the other is closed (remark that if $H$ and $K$ are closed, $f$ and $g$ can have no cluster point other than $e$ when $x$ tends to $e$ in $G$).