Continuous Mapping of Separation

Theorem
Let $T$ and $T'$ be topological spaces.

Let $A \mid B$ be a separation of $T$.

Let $f: T \to T'$ be a mapping such that the restrictions $f \restriction_A$ and $f \restriction_B$ are both continuous.

Then $f$ is continuous on the whole of $T$.

Proof
Follows directly from Pasting Lemma for Union of Open Sets.