Theorem Let ${B}=\{x\in\Reals^n: \norm{x}\le 1\}$ be the closed unit ball in $\Reals^n$ . Any continuous function $f: {B}\to{B}$ has a fixed point.

Notes

Shape is not important

The theorem also applies to anything homeomorphic to a closed disk, of course. In particular, we can replace ${B}$ in the formulation with a square or a triangle.

Compactness counts (a)

The theorem is not true if we drop a point from the interior of ${B}$ . For example, the map $f(\vec{x})=\frac{1}{2}\vec{x}$ has the single fixed point at $0$ ; dropping it from the domain yields a map with no fixed points.

Compactness counts (b)

The theorem is not true for an open disk. For instance, the map $f(\vec{x})=\frac{1}{2}\vec{x}+(\frac{1}{2},0,\ldots,0)$ has its single fixed point on the boundary of ${B}$ .