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I got this questions:

Prove or disprove by a counterexample the following statements:

Let $f:\mathbb{R}\to\mathbb{R}$ be a function that is integrable on every closed interval and let $F(x)=\int_0^x f(t)\;dt$. (1) If $f$ is bounded on $\mathbb{R}$, Then $F$ is uniformly continuous on $\mathbb{R}$.
(2) If $f$ is continuous on $\mathbb{R}$, Then $F$ is uniformly continuous on $\mathbb{R}$.

Some hints will be helpful. Thanks.

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  • $\begingroup$ Since this is a site that encourages learning, you will get much more help if you show us what you have already done. Could you edit your question with your thoughts and ideas? $\endgroup$ – 5xum Jun 6 '14 at 11:40
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Hint: Only one of the two statements is true. The other can be disproven using very simple functions.

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  • $\begingroup$ Thanks for the hint. I took $f(x)=2x$ and showed that $F(x)=x^2$ which is not uniformly continuous on $\mathbb{R}$ and (2) disprooved. $\endgroup$ – MathNerd Jun 6 '14 at 12:11

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