Brouwer's Fixed Point Theorem/One-Dimensional Version/Proof by Intermediate Value Theorem
- $\exists \xi \in \closedint a b: \map f \xi = \xi$
Thus $\map f a \ge a$ and $\map f b \le b$.
Let us define the real function $g: \closedint a b \to \R$ by $\map g x = \map f x - x$.
But $\map g a \ge 0$ and $\map g b \le 0$.
By the Intermediate Value Theorem, $\exists \xi: \map g \xi = 0$.
Thus $\map f \xi = \xi$.
Source of Name
This entry was named for Luitzen Egbertus Jan Brouwer.