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Let $G$ be a polish group, $H$ be an open subgroup of $G$ and $X$ be any metric space on which $H$ act continuously. Let $f:G\longrightarrow X$ such that $\forall h\in H$ and $\forall g\in G$, $f(gh)=h^{-1}f(g)$. I want to show that $f$ is continuous. Thank for any help

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You probably solved your problem already since the question was asked quite a long time ago. Just in case, here is a proof.

The group $G$ is a disjoint union of $H$-cosets $gH$. Since these cosets are open, it is enough to show that the restriction of $f$ to any fixed coset $g_0H$ is continuous. But this is clear since $f(g_0h)=h^{-1} f(g_0)$ for every $h\in H$ and the action is continuous.

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