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.