Continuity Test using Sub-Basis/Proof 2

Proof
Let $\tau$ be the final topology on $X_2$ with respect to $f$.

By hypothesis, $\mathcal S \subseteq \tau$.

By Synthetic Sub-Basis and Analytic Sub-Basis are Compatible, we have that $\tau_2$ is the topology generated by the synthetic sub-basis $\mathcal S$.

By the definition of the generated topology, we have $\tau_2 \subseteq \tau$.

By the definition of the final topology, it follows that $f$ is continuous.