I was thinking in The following situation. Let $X$ be a topological space and $Y=X/\sim$ with quotient map $q:X\to Y$. Let $f:Y\to Y$ be a function. Is there an easy way to prove that $f$ is continous provided that $\tilde{f}:X\to X$, a function that makes the same assignment as $f$ is continous $q\circ \tilde{f} = f\circ q$.
I know that, by The universal property of quotient maps, a map $f:Y\to Y$ is continous iff $f\circ q$ but I need to go further. Any suggestion? Is there a general result?