Let $f : X \to Y$ be continuous. Let $R$ be equivalence relation on $X$ such that $xRy$ only if $f(x) = f(y)$ and $S$ be equivalence relation on $Y$.
Let $g: X/R \to Y$ where $g([x]_R) := f(x)$ and $h: X \to Y/S$ where $h(x) := [f(x)]_S$.
Are functions $g$ and $h$ continuous? I must admit I don't have an idea how to approach such problem. Any insights are welcome, thanks in advance for your help.
$g$ is continuous as $g \circ q_R = f$ and $f$ is continuous, by the universal property of quotients.
$h$ is just the composition $q_S \circ f$ so continuous as a composition of continuous maps. Note that $f$ does not have any relation to $S$, so $h$ is less likely to be a useful map.