1. The problem statement, all variables and given/known data Let f : [0; 1] --> R be continuous on [0, 1], and assume that the range of f is contained in [0; 1]. Prove that there exists an x in [0, 1] satisfying f(x) = x. 2. Relevant equations 3. The attempt at a solution Well i am almost positive I need to use the intermediate value theorem. First I could claim that either f(0)>x>f(1) or f(0)<x<f(1). where x is a value in (0,1) Not too sure what to do, i think the key is something to do with the range of f being contained in [0,1], but any help would be great!