Pasting Lemma/Corollary 1

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Theorem

Let $X$ and $Y$ be topological spaces.

Let $A$ and $B$ be closed in $X$.

Let $f : A \to Y$ and $g : B \to Y$ be continuous mappings that agree on $A \cap B$.


Then the mapping $f \cup g : A \cup B \to Y$ is continuous.


Proof