Suppose we have a continuous function f that is on the closed interval [0,1] and has a range that's contained in [0,1] as well. Prove f has to have a fixed point (ie: show for at least 1 value of .
We are done if or
What about if and .
Let , then we know that is continuous and further . By the Intermediate Value Theorem (IVP), there exists a with . That is, . And hence the proof is complete.