Are the following statements True or False?

  1. If $ f : (0, \infty) \rightarrow \mathbb{R}$ is uniformly continuous then $\lim_{x \rightarrow 0} f(x)$ exists.

  2. If $ f : (0, \infty) \rightarrow \mathbb{R}$ is uniformly continuous then $\lim_{x \rightarrow \infty} f(x)$ exists.

For part 1, this is what I tried : consider $\{x_n\}$, $\{y_n\}$ be two Cauchy sequences converging to 0. Then by uniform continuity $\{f(x_n)\}$ and $\{f(y_n)\}$ are Cauchy and convergent to the same point $l = \lim_{n \rightarrow \infty} f(x_n)$. To show $l = f(\lim_{n \rightarrow \infty} x_n) = f(0)$, which is satisfied as f is continuous on $(0, \infty)$ hence the limits can be exchanged. Hence $\lim_{x \rightarrow 0} f(x)$ exists.

For the second part, by uniform continuity on $[b, \infty)$, $b$ sufficiently large, given $\epsilon > 0$, there exists $\delta > 0$ such that whenever $|x-y| < \delta$, $|f(x)- f(y)| < \epsilon$. Consider a sequence $x_n \rightarrow \infty$. I am stuck here.

Could you please check my ideas and help me finish the problems? Thank you.


The first assertion is indeed true. The second one is false. Take the sine function, for instance.

  • $\begingroup$ oh thank you, such a simple example, i didn't think of it! $\endgroup$
    – ymir
    Mar 19 '21 at 17:09
  • $\begingroup$ If my answer was useful, perhaps that you could mark it as the accepted one. $\endgroup$ Mar 19 '21 at 18:11
  • $\begingroup$ It's actually possible to have $f(\{M,\infty)\})=\mathbb R$ for every $M>0.$ $\endgroup$
    – zhw.
    Mar 19 '21 at 20:22
  • $\begingroup$ @JoséCarlosSantos You might want to wait a day or so before saying this. $\endgroup$
    – zhw.
    Mar 19 '21 at 20:23

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