Brouwer fixed-point theorem

A fixed point of a function f is a point x for which f(x) = x. Brouwer's theorem says that if f is continuous and maps a closed ball in a finite-dimensional Euclidean space into that same ball, then at least one such point exists. The result also applies to sets homeomorphic to a closed ball, including compact convex subsets of Euclidean space. Continuity and the shape of the domain matter: the conclusion is not a claim that every self-map of every set has a fixed point.

On a closed interval, a continuous function that maps the interval into itself must meet the identity function somewhere, giving a fixed point; this one-dimensional case is closely related to the intermediate value theorem. In higher dimensions the analogous claim is less visually obvious, but the theorem extends it to balls. For example, a continuous transformation that moves points around within a filled disk still leaves at least one point in place, though the theorem does not say where it is.

Dutch mathematician L. E. J. Brouwer stated and proved the theorem in 1912, within the development of algebraic topology. Fixed-point theorems have become useful in many existence proofs: in economics, for example, suitable fixed-point results help establish the existence of competitive equilibria and Nash equilibria. The precise theorem and assumptions used depend on the mathematical model; Brouwer's theorem alone does not ensure that an equilibrium is unique, stable, or easy to find.

Brouwer's theorem is an existence result. It does not generally supply an explicit formula, a computational procedure, or uniqueness. Those further conclusions require additional structure or methods. Its hypotheses are also essential: changing the domain so it is not compact or ball-like, or allowing a discontinuous map, can invalidate the conclusion.