Brouwer fixed-point theorem

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) 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}} .