Pasting Lemma/Corollary 2
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Theorem
Let $X$ and $Y$ be topological spaces.
Let $\mathcal A = \left\{ {A_i: i \in I} \right\}$ be a set of sets that are open in $X$.
Let $f: \bigcup \mathcal A \to Y$ be a mapping such that:
- $\forall i \in I : f \restriction A_i$ is continuous
Then $f$ is continuous on $\bigcup \mathcal A$.
Proof
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