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1. The pasting lemma: $A$ and $B$ are closed sets. If $f:A \to Y$ and $g: B \to Y$ are both continuous, and $f(x)=g(x)$ for all $x \in A \cap B$, then $h: (A \cup B) \to Y$ defined by $h(x)=f(x)$ for $x \in A$ and $h(x)=g(x)$ for $x \in B$ is also continuous.
2. Insight: Continuity (of $f: X \to Y$) does not only depend on the function itself, but also depend on the topologies on both $X$ and $Y$. Indeed, with weaker topology on $X$ or stronger topology on $Y$, there are less continuous functions from $X$ to $Y$; on the contrast, with stronger topology on  $X$ or weaker topology on $Y$, there are more continuous functions from $X$ to $Y$.