Continuous Mapping of Separation

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

Let $$A | B$$ be a partition 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 Continuity from Union of Restrictions.