If g is a continuous function g(x) element [a, b] for all x element [a, b], then g has a fixed point in [a, b]. This can be proven by supposing that g(a)>=a g(b)<=b g(a) - a>=0 g(b) - b<=0. Since g is continuous, the intermediate value theorem guarantees that there exists a c element [a, b] such that g(c) - c = 0, so there must exist a c such that g(c) = c, so there must exist a fixed point element [a, b].

Banach fixed point theorem | Brouwer fixed point theorem | hairy ball theorem | Kakutani's fixed point theorem | Lefschetz fixed point theorem | Lefschetz trace formula | map fixed point | Poincaré-Birkhoff fixed point theorem | Schauder fixed point theorem