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I'm supposed to show the equality of the 3 following statements:

$f:]a,b[ \rightarrow \mathbb{R}$ is continous with $-\infty < a < b < \infty$

1) $f$ is uniformly continous

2) A continuous function $g:[a,b] \rightarrow \mathbb{R}$ with $f(x) = g(x)$ $\forall x\epsilon ]a,b[$ exists

3) The limits $\lim_{x\rightarrow a}f(x)$ and $\lim_{x\rightarrow b}f(x)$ exist

I also need to show that for $a=-\infty $ and $b=\infty $, $3\Rightarrow 1$ still applies but $1\Rightarrow 3$ generally does not

Ayelle
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  • I think the best way to do it is $3 \implies 2 \implies 1 \implies 3$. See if you can do it this way and ask specific questions if you get stuck. – Jon Warneke Jan 25 '16 at 00:28
  • For the second question, let $\varepsilon >0$, and take a closed interval $[-N,N]$ with $N$ large enough so that $f(-N)$ is within $\varepsilon$ of $\lim_{x \ to -\infty} f(x)$ and $f(N)$ is within $\varepsilon$ of $\lim_{x \ to \infty} f(x)$. (Why does such $N$ exist?) Then we can use the uniform continuity of $f$ on $[-N,N]$ and the closeness of $f$ to is limits away from $[-N,N]$ to establish uniform continuity on all of $\mathbb R$. Again, try it and ask specific questions if you get stuck. – Jon Warneke Jan 25 '16 at 00:40

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