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Does uniform continuity imply boundedness?

I know this question has been asked many times on this site. However, I found different answers from different people.

Uniform continuity and boundedness

This link seems to suggest that it does.

However, f(x) = x seems to be a counter example.

So which one is correct? Or is the link proving other things which I am not aware of.

Thanks

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user10024395
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  • The hypothesis given in that link is "If $f$ is uniformly continuous on a bounded interval $I$". $f(x) = x$ is uniformly continuous and bounded on any bounded interval. – Umberto P. Apr 24 '14 at 14:39
  • That's depende on the domain of your function. If you are considering a bounded interval obvious yes – rlartiga Apr 24 '14 at 14:39
  • This link seems to suggest that it does "when the domain of $f$ is a bounded interval." As long as you're considering $f:x\mapsto x$ on a bounded interval, it's bounded. – Sugata Adhya Apr 24 '14 at 14:39

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If $f$ is uniformly continuous on a bounded set, then $f$ is bounded.

But if the domain of $f$ is not bounded, then $f$ may be not bounded too, as you have proved with the function $f(x)=x$.

ajotatxe
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