# Brouwer's Fixed Point Theorem

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## Theorem

### One-Dimensional Version

Let $f: \closedint a b \to \closedint a b$ be a real function which is continuous on the closed interval $\closedint a b$.

Then:

- $\exists \xi \in \closedint a b: \map f \xi = \xi$

That is, a continuous real function from a closed real interval to itself fixes some point of that interval.

### Smooth Mapping

A smooth mapping $f$ of the closed unit ball $B^n \subset \R^n$ into itself has a fixed point:

- $\forall f \in \map {C^\infty} {B^n \to B^n}: \exists x \in B^n: \map f x = x$

### General Case

A continuous mapping $f$ of the closed unit ball ${B^n}^- \subset \R^n$ into itself has a fixed point:

- $\forall f \in \map {C^0} { {B^n}^- \to {B^n}^-} : \exists x \in {B^n}^- : \map f x = x$

## Also known as

**Brouwer's Fixed Point Theorem** is also known just as **Brouwer's Theorem**.

## Also see

## Source of Name

This entry was named for Luitzen Egbertus Jan Brouwer.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Brouwer's theorem** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Brouwer's theorem**