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Let $G$ be a topological group acts on the topological space $X$, for an elememt $g\in G$, let's define the map $f:X\rightarrow X, f(x)=g*x$.

I am trying to find if $f$ is continuous?

my best thanks

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Without further condition on the group action, $f_g$ is not necessarily continuous. Usually, with topological groups, you specify that the group action: $\phi:G\times X\rightarrow X$ is continuous. Then $f_g(x)=\phi(g,x)$ is continous. –  Thomas Andrews Oct 19 '12 at 0:09
    
yes, actually $\phi:G\times X \rightarrow X$ is continuous, but how can we prove that $f_{g}:X\rightarrow X$ defined above is continuous –  Kamal Oct 19 '12 at 0:16
3  
Because $h_g:X\to G\times X$ defined as $h_g(x)=(g,x)$ is continuous. And $f_g=\phi \circ h_g$ –  Thomas Andrews Oct 19 '12 at 0:22
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