Continuous Mapping of Partition

Theorem

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

Let $A \mid B$ 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.

$\blacksquare$