Using the definition of the derivative to prove a constant function I am presented with the following task:
"Let $f$ be a function defined on the interval $I$. All we know about $f$ is that there is a constant $K$ such that 
$$|f(a) - f(b)| \leq K|a-b|^2$$
for all $a, b \in I$. Show that $f$ is constant on $I.$ (Hint: calculate the derivative using the definition of the derivative first.)"
I am utterly confused. The Mean-Value Theorem crosses my mind while looking at the equation. I believe that I am supposed to somehow prove that $f(a) - f(b) = 0$ for all $a, b$, thus proving that the function is constant on $I$, but I'm having a hard time seeing how to make any progress in that direction.

Take $\;a,x\in I\;$ , so we're given :
$$-K|x-a|\le\frac{f(x)-f(a)}{x-a}\le K|x-a|$$
pass now to the limit of the above when $\;x\to a\;$ and use the squeeze theorem. You get the middle term's limit exists and equals zero, and since that term's limit defines $\;f'(a)\;$ you've proved both that the derivative at $\;a\;$ exists and equals zero...
