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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.

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    $\begingroup$ @user539887 thanks. edited the post $\endgroup$ – vidyarthi Dec 10 '18 at 7:30
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$f(x)=x$ is a counterexample for the first part. Lipschitz functions need not be differentiable at every point but they are differentiable almost everywhere.

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No, consider the function $f(x)=|x|$ which is Lipschitz in $\mathbb{R}$ (with constant $1$), not differentiable at $0$, and $$\lim_{x\to\pm\infty}\frac{f(x)}{|x|}=\pm 1.$$

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  • $\begingroup$ @vidyarthi Are you still interested in this question? $\endgroup$ – Robert Z Dec 10 '18 at 9:28

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