Suppose f : R ------> R, and suppose that $\displaystyle \vert f(x) - f(y) \vert \leq (x - y)^2 $ for all x, y in R.

Prove that f is a constant.

I tried solving this problem using the Mean Value Theorem, and showing that f'(xo) is always zero.

$\displaystyle \vert f'(xo) \vert = \vert \frac{f(x) - f(y)}{x -y} \vert \leq \frac{(x-y)^2}{(x-y)} \leq (x-y) \leq \epsilon $

Now, is there an argument I can use to let (x-y) be less than epsilon? Is there something that I'm overlooking?

Any insight would be welcomed,

Thanks