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Let $f :(0, 1)→ R$ be continuous. Pick out the statements which imply that $f$ is uniformly continuous.

a. $|f(x) − f(y)| ≤ \sqrt{|x − y|}, \text{ for all }x, y \in [0, 1].$

b. $f\left(\frac{1}{n}\right)\rightarrow \frac{1}{2}$ and $f\left(\frac{1}{n^2}\right)\rightarrow \frac{1}{4}.$

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  • $\begingroup$ It is even discontinuous in $0$ about the $f$ in $(b)$. And $f$ in $(a)$ satisfies Lipschitz Condition so it is uniformly continuous. $\endgroup$
    – gaoxinge
    Commented Jan 22, 2014 at 8:45
  • $\begingroup$ Domain of function is open (0,1) $\endgroup$
    – user121418
    Commented Jan 22, 2014 at 8:47
  • $\begingroup$ Lemma. $f$ is uniformly in $(a,b)$ $\Longleftrightarrow$ the limits of $a,b$ are exist. Then you can extend the function on whole $[a,b]$. $\endgroup$
    – gaoxinge
    Commented Jan 22, 2014 at 8:51
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    $\begingroup$ If |f(x)−f(y)|≤k.|x−y|, then f satisfies lipscitz condition, in a it is not satisfied- gaoxinge $\endgroup$
    – user121418
    Commented Jan 22, 2014 at 8:52
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    $\begingroup$ (b) can't even happen. If $f(1/n) \to 1/2$, then for $n$ big enough $f(1/n) > 3/8$. In particular for $n$ big enough $f(1/n^2) > 3/8$, and so $f(1/n^2)$ can't converge to $1/4$. $\endgroup$ Commented Jan 22, 2014 at 8:56

2 Answers 2

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Notice this

$$ |f(x) − f(y)| ≤ \sqrt{|x − y|} < \epsilon \implies |x-y|< \epsilon^2 = \delta. $$

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  • $\begingroup$ what about option b? $\endgroup$
    – user121418
    Commented Jan 22, 2014 at 8:45
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    $\begingroup$ impossible since $n^2$ is a subsequence of $n$ $\endgroup$
    – Blah
    Commented Jan 22, 2014 at 8:48
  • $\begingroup$ Mhenni Benghorbal can you give me a conterexample or proper justification for case b in your answer? $\endgroup$
    – user121418
    Commented Jan 22, 2014 at 9:01
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For option $(b)$ does not implies that uniform convergence. Take $a_n=\frac{1}{n}$ and $b_n=\frac{1}{n^2}$.Here $a_n-b_n \rightarrow 0$ but $f(a_n)-f(b_n)\nrightarrow 0$ . Hence (b) does not implies uniform continuity.

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  • $\begingroup$ please read the question Murugan $\endgroup$
    – user121418
    Commented Jan 22, 2014 at 9:03
  • $\begingroup$ See! suppose (b) implies f is uniformly continuous. Then if we take a seqn $a_n $ and $b_n$ as the above. By uniform continuity $f(a_n)-f(b_n)$ goes to zero. so contradiction. $\endgroup$
    – Murugan
    Commented Jan 22, 2014 at 9:06
  • $\begingroup$ sorry my mistake. thank you. $\endgroup$
    – user121418
    Commented Jan 22, 2014 at 9:51
  • $\begingroup$ It seems to me that "does not imply" is, technically, the wrong answer to part b) in view of nik's comment above. E.g. doesn't $1 = 2$ imply that $f$ is uniformly continuous? $\endgroup$ Commented Jan 22, 2014 at 18:36

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