Pasting Lemma/Corollary 2

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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 $\bigcup \mathcal A$.