## Fixed Point Theorem

If is a continuous function For All , then has a Fixed Point in . This can be proven by noting that

Since is continuous, the Intermediate Value Theorem guarantees that there exists a such that

so there must exist a such that

so there must exist a Fixed Point .

See also Banach Fixed Point Theorem, Brouwer Fixed Point Theorem, Kakutani's Fixed Point Theorem, Lefshetz Fixed Point Formula, Lefshetz Trace Formula, Poincaré-Birkhoff Fixed Point Theorem, Schauder Fixed Point Theorem