1

Let $ f: (a,b) \rightarrow \mathbb{R}$ continuous. Show that $f$ is uniformly continuous if and only if the limits $\lim_{x \rightarrow a^+} f(x)$ and $\lim_{x \rightarrow b^-} f(x)$ exist in $\mathbb{R}$

I chose an $\epsilon$ = 1 and attempted using the definition of uniform continuity. I was hoping for a contradiction with $|x-y| < \epsilon$ but can't find one.

user381164
  • 207
  • 1
    If the limits exist at both ends, you can add those limits to your function and form $g=f$ on $(a,b)$ and $g(a)=\lim_{x\to a+} f(a)$ and similarly for $g(b)$. Then $g$ is continuous on $[a,b]$ so it is uniformly continuous. – user160738 Nov 04 '16 at 17:59
  • As for the other direction you can use this :http://math.stackexchange.com/questions/569036/how-to-prove-cauchy-criterion-for-limits – user160738 Nov 04 '16 at 18:15
  • hmm i'm not seeing it. Could I also use this: http://math.stackexchange.com/questions/75491/how-does-the-existence-of-a-limit-imply-that-a-function-is-uniformly-continuous?rq=1 ? – user381164 Nov 07 '16 at 19:09
  • you have essentially two different cases: both $a,b$ are finite, or one or both of them are infinite. $a,b$ are finite then what I said makes sense, if one of the say $b$ is infnite then you can use that, but really it's reverse direction that's non trivial – user160738 Nov 08 '16 at 12:03

0 Answers0