Brouwer fixed-point theorem

Theorem in topology

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. Brouwer. It states that for any continuous function f {\displaystyle f} mapping a nonempty compact convex set to itself, there is a point x 0 {\displaystyle x_{0}} such that f ( x 0) = x 0 {\displaystyle f(x_{0})=x_{0}} . The simplest forms of Brouwer's theorem are for continuous functions f {\displaystyle f} from a closed interval I {\displaystyle I} in the real numbers to itself or from a closed disk D {\displaystyle D} to itself.

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