The question:

Suppose that f is continuous on [0,1] and that Range(f) is a subset of [0,1]. By using g(x) = f(x) - x, prove that there is a real number c in [0,1] such that f(c) = c.

I'm not sure where to start. I know that the intermediate value theorem applies, since we're given a function g(x) which is continuous on the interval [0,1]. However, I don't know how to use this to prove that f(c) = c.

Any help would be great.