Brouwer fixed-point theorem

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function ${\displaystyle f}$ mapping a compact convex set into itself there is a point ${\displaystyle x_{0}}$ such that ${\displaystyle f(x_{0})=x_{0}}$. The simplest forms of Brouwer's theorem are for continuous functions ${\displaystyle f}$ from a closed interval ${\displaystyle I}$ in the real numbers to itself or from a closed disk ${\displaystyle D}$ to itself. A more general form than the latter is for continuous functions from a convex compact subset ${\displaystyle K}$ of Euclidean space to itself.

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