Pasting lemma

In topology, the pasting or gluing lemma, and sometimes the gluing rule, is an important result which says that two continuous functions can be "glued together" to create another continuous function. The lemma is implicit in the use of piecewise functions. For example, in the book Topology and Groupoids, where the condition given for the statement below is that ${\displaystyle A\setminus B\subseteq \operatorname {Int} A}$ and ${\displaystyle B\setminus A\subseteq \operatorname {Int} B.}$

The pasting lemma is crucial to the construction of the fundamental group and fundamental groupoid of a topological space; it allows one to concatenate paths to create a new path.