Analysis of Differentiable Functions Suppose that $f : \Bbb{R} \to \Bbb{R}$ is a function such that $|f(x)− f(y)| ≤ |x−y|^2$ for all $x$ and $y$. Show that $f (x) = C$ for some constant $C$. 
Hint: Show that $f$ is differentiable at all points and compute the derivative
I confused as to what I use as the function in order to show that $f$ is differentiable at all points

Hint: Let $y=x+h$, then you have
$$abs\left( \frac{f(x+h)-f(x)}{h}\right)\le |h|,$$ so that as $h \to 0$ you get that $f$ is differentiable. Maybe now you can use differentiability of $f$ to finish.
Actually once you know it's differentiable, the same inequality above shows the derivative is $0$, so not really more work, just to explain it on your write-up.
