Let $X$ and $Y$ be topological spaces, and let $f: X \to Y$ be a function.
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1. **(Open pasting)** If $\{U_\alpha\}_{\alpha \in I}$ is an open cover of $X$ and $f|_{U_\alpha}: U_\alpha \to Y$ is continuous for each $\alpha \in I$, then $f$ is continuous.
2. **(Closed pasting)** If $X = C_1 \cup C_2$ where $C_1, C_2$ are closed, and both $f|_{C_1}$ and $f|_{C_2}$ are continuous, then $f$ is continuous.