Let $f$ be a function with continuous derivatives on the real axis $\mathbb R$ which satisfies$\forall x\in\mathbb R\forall y\in\mathbb R(|f(x)-f(y)|\geq m|x-y|)$
where $m > 0$ and is a constant. Prove:
(1) If $E$ is a zero-measurable set, then $f(E)$ is a zero-measurable set
(2) If $E$ is a measurable set, then $f(E)$ is a measurable set
I have tried as much as I can, but I still can't find a way to solve this problem. Feel free to leave your thoughts and answers :)
