Solution
= Solution
If $g$ is <continuous>, so is the composition $g\circ p$. Conversely, assume the composition is <continuous>. For every <open set> $V\subseteq Y$,
$$
p^{-1}(g^{-1}(V))=(g\circ p)^{-1}(V)
$$
is <open> in $X$. The quotient definition then says that $g^{-1}(V)$ is <open> in $X/\mathcal R$. Hence $g$ is <continuous> and
$$
\boxed{g\text{ continuous}\iff g\circ p\text{ continuous}.}
$$
This is the <universal property of the quotient topology>.