f: is continuous and bounded then f has a fixed point.
Discussion: Consider the function g(x) = f(x) – x. Since f(x) and x are continuous then g(x) must be continuous. Since f(x) is bounded then there exists a M such that f(x) < M.
Now as the codomain is . Thus . Also g(M) must be negative as f(M) < M. Since g(x) is continuous, by Intermediate Value Property of Continuous Functions g(x) must attain the value of 0 somewhere between x = 0 to x = M. Suppose that value is c.
Hence g(c) = f(c) – c = 0 or f(c) = c.