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Let $f(x)$ be a Lipschitz continuous gradient function, that is $$ \|f'(x)-f'(y)\| \leq \alpha \|x-y\| $$ where $\|\cdot\| $ is Euclidean norm and $x,y \in \mathbb{R}^n$. Can we prove that the function is a Lipschitz function, that is $$ |f(x)-f(y)| \leq L \|x-y\| $$ If so, is there any relationship between $\alpha$ and $L$?

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  • $\begingroup$ Did you try to see what happens when $n=1$? $\endgroup$ – Umberto P. Aug 28 '18 at 23:05
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    $\begingroup$ Try with $f(x) = x^2$. $f'$ is globally Lipschitz, $f$ is not. $\endgroup$ – copper.hat Aug 28 '18 at 23:14

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