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.
I think there's a theorem in my book that I'm allowed to use. I think it's something along the lines of:
With a continuous function the two points are connected, so the function itself must cross the axis, thus resulting in an intermediate value.
Is that correct? Not sure if I explained it right.