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.

Suppose for example that $0\in I$ and $1\in I$.
Then
$$
|f(1)-f(0)|=\bigg|\sum_{k=0}^{n-1} f(\frac{k+1}{n})-f(\frac{k}{n})\bigg|
\leq \sum_{k=0}^{n-1} \big|f(\frac{k+1}{n})-f(\frac{k}{n})\big|
\leq \sum_{k=0}^{n-1} K\frac{1}{n^2}=\frac{K}{n}
$$
Since this holds for any $n$, we deduce $f(1)=f(0)$.
You can do this for any pair other than $(0,1)$.
