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Let $g$ be a continuously differentiable function on $\mathbb{R}$. Then, is the function $\int_0^xg\ dt$ uniformly continuous on $\mathbb{R}$? Is it also lipschitz continuous? I think it is uniformly continuous, by using the fundamental theorem of calculus(or mean value theorem), but am unsure about the rigour of proof required. Any hints. Thanks beforehand.

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Neither. $F(x)=x^2$ - or any polynomial of degree $\ge2$, for that matter - is not UC on $\Bbb R$, so $g(x)=2x$ is a counterexample.

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  • $\begingroup$ thanks, was so silly of me! $\endgroup$ – vidyarthi Sep 16 '17 at 7:26

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