The concept of continuity is fundamental in large parts of contemporary mathematics. In the nineteenth century, precise de nitions of continuity were formulated for functions of a real or complex variable, enabling mathematicians to produce rigorous proofs of fundamental theorems of real and complex analysis, such as the Intermediate Value Theorem, Taylor's Theorem, the Fundamental Theorem of Calculus, and Cauchy's Theorem.

In the early years of the Twentieth Century, the concept of continuity was generalized so as to be applicable to functions between metric spaces,  and subsequently to functions between topological spaces.