Let f be continuous on the interval [0,1] to Reals and such that f(0) = f(1). Prove that there exists a point c in [0,1/2] such that f(c) = f(c + (1/2)).

[Hint: Consider g(x) = f(x) - f(x + (1/2))]

I'm not really sure how to go about proving this. Any help would be much appreciated.