 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 $W\subseteq Z$,is open in $X$. The quotient topology criterion therefore says $g^{-1}(W)$ is open in $X/\mathcal{R}$. This proves continuity of $g$.This is the universal property of the quotient topology.