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The Pasting Lemma
Definition: Let X=A union B, where A and B are closed in X. Let f: A -> Y and g: B -> Y be continuous. If f(x)=g(x) for every x in A intersect B, then f and g combine to give a continuous function h: X -> Y, defined by setting h(x)=f(x) if x is in A, and h(x)=g(x) if x is in B.
Source: Topology (second edition) by James R. Munkres