= Solution
If $g$ is <continuous>, then so is $g\circ p$, as a <composition of continuous functions>. Conversely, suppose $g\circ p$ is continuous. For an <open subset> $W\subseteq Z$,
$$
p^{-1}(g^{-1}(W))=(g\circ p)^{-1}(W)
$$
is open in $X$. The <quotient topology> criterion therefore says $g^{-1}(W)$ is open in $X/\mathcal R$. This proves continuity of $g$.
$$
\boxed{g\text{ continuous}\iff g\circ p\text{ continuous}.}
$$
This is the <universal property of the quotient topology>.
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