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Let $I$ be a bounded interval. Is there a function $f$, which is continuous and bounded on $I$, but it's not uniformly continuous on $I$ ?

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  • $\begingroup$ Can $I$ be open? $\endgroup$
    – zhw.
    Aug 21, 2016 at 1:49
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    $\begingroup$ Yes. I think it must be, according to the Heine-Cantor theorem $\endgroup$
    – Patrik Bak
    Aug 21, 2016 at 1:56

1 Answer 1

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$\sin \frac 1x$ on $(0,1]$

That function is bounded, continuous, but is not uniformly continuous.

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  • $\begingroup$ How can you prove that the function is not uniformly continuous? $\endgroup$
    – Patrik Bak
    Aug 23, 2016 at 1:18

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