Real Analysis on Functions with Lipschitz Derivatives

Question

I saw a wired argument somewhere and thought maybe this would be a good practice and somehow I can't wrap my head around it:

Let $f(x)$ be Lipschitz and have Lipschitz derivatives everywhere.

If we know

$$|f'(x_1)-f'(x_2)|>\epsilon$$

can we conclude

$$|x_1-x_2|>\epsilon^2$$

Answer 1

The claim is true, but only for sufficiently small $\epsilon$. We'll work with the contrapositive. If $|x-x_2| \leq \epsilon^2$, then since $f'$ is Lipschitz, we get that $|f'(x_1)-f'(x_2)| \leq M \epsilon^2$ for some constant $M>0$. Now, note that $M\epsilon^2 \leq \epsilon$ provided $\epsilon \leq \frac{1}{M}$.