Corollary 2

Let $X, Y$ be two topological spaces. In order that a mapping $f : X \to Y$ should be continuous it is necessary and sufficient that the mapping $g : x \to (x, f(x))$ is a homeomorphism of $X$ onto the graph $G$ of $f$ (considered as a subspace of the product space $X \times Y$).

Since $f = \mathrm{pr}_2 \circ g$, the condition is sufficient. It is also necessary, for if $f$ is continuous, then $g$ is bijective and continuous (Proposition 1) and the inverse of $g$ is the restriction of $\mathrm{pr}_1$ to $G$, which is continuous (cf. Set Theory, Chapter IV, § 2, no. 4, criterion CST 17).