Functions

A rule assigning exactly one output to each valid input, the single most pervasive organizing idea in mathematics above arithmetic itself. Gottfried Leibniz coined the term function in 1673 for a quantity depending on a curve, but the modern, fully general notion, and the now-universal notation f(x) for the value a function f assigns to an input x, is due to Leonhard Euler, who developed and popularized it across his extraordinarily productive career, especially in his 1748 Introductio in analysin infinitorum. The definition has itself been generalized more than once since; the fully set-theoretic definition, a function as a particular kind of set of ordered pairs, is a twentieth century refinement built on Cantor's set theory.