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