On page 341 of Follad's Real Analysis 2nd edition, it is written that "it is easy to see that every Borel measurable function on G is constant on the cosets of H" where $G$ is a topological group and $H$ is the closure of $\{e\}$ where $e$ the identity element of $G$.
With the understanding that by a "Borel-measurable map" we mean a complex or real-valued function defined on $G$, I can see that the statement is true for all continuous functions, but I'm not sure why this must be true for all Borel-measurable maps.
Any help would be appreciated. Thank you.