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.
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!