Is a bounded and continuous function uniformly continuous?

Question

$f\colon(-1,1)\rightarrow \mathbb{R}$ is bounded and continuous does it mean that $f$ is uniformly continuous?

Well, $f(x)=x\sin(1/x)$ does the job for counterexample? Please help!

See also http://math.stackexchange.com/questions/1315555/an-example-of-a-bounded-continuous-function-on-0-1-that-is-not-uniformly-co — Martin Sleziak, Feb 09 '16 at 16:24

score 19 · Accepted Answer · answered Oct 25 '12 at 10:33

19

You're close: $$\sin\frac{1}{x+1}$$ is a counterexample to the statement.

answered Oct 25 '12 at 10:33

asdf

206
2
2

score 9 · Answer 2 · edited Aug 18 '23 at 06:17

9

For continuity to lead to uniform continuity, domain has to be compact, and as you can see the domain is not compact here. Also, rightly $f(x)=\sin(\frac{1}{x+1}) $ serves as a counterexample.

edited Aug 18 '23 at 06:17

Sarvesh Ravichandran Iyer

76,105

answered Oct 25 '12 at 11:14

Vishesh

2,928

10

$\sin(e^x)$ is uniformly continuous on $(-1,1)$. – commenter Oct 25 '12 at 11:17
1

Yes it is. Thanks. Don't know what I was thinking. – Vishesh Oct 25 '12 at 11:18

score 4 · Answer 3 · edited Apr 25 '13 at 16:06

4

$\sin(x^2)$ is also a nice example and it's happening because it's not periodic.

edited Apr 25 '13 at 16:06

Rudy the Reindeer

46,359

answered Apr 25 '13 at 14:51

Arghya Ghosh

89

4

This function is continuous on $[-1,1]$, so it is uniformly continuous there. A fortiori on $(-1,1)$. – Julien Apr 25 '13 at 16:37
Over the real line, this is, not on any bounded interval. – Pedro Apr 25 '13 at 16:39
1

This answer is wrong. Although its a nice counterexample to show that a bounded function on a closed, but unbounded interval is not necessarily uniformly continuous. – jodag Feb 28 '18 at 00:13

Is a bounded and continuous function uniformly continuous?

3 Answers3

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