# Lipschitz functions are $o(|x|)$?

Consider a Lipschitz function from $$\mathbb{R}\to\mathbb{R}$$. Can we say that $$\lim_{x\to\infty}\frac{f(x)}{|x|}=0$$. Can we also say that $$f$$ is differentiable. Continuity is quite evident. But linear order is hard to come by. Any hints? Thanks beforehand.

• @user539887 thanks. edited the post – vidyarthi Dec 10 '18 at 7:30

$$f(x)=x$$ is a counterexample for the first part. Lipschitz functions need not be differentiable at every point but they are differentiable almost everywhere.
No, consider the function $$f(x)=|x|$$ which is Lipschitz in $$\mathbb{R}$$ (with constant $$1$$), not differentiable at $$0$$, and $$\lim_{x\to\pm\infty}\frac{f(x)}{|x|}=\pm 1.$$