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Please prove whether the function $$f(x) = \frac{1}{\sin(x)}$$ is uniformly continuous or not on $(0,1)$.

I found online where $\sin(1/x)$ is uniformly continuous but not this one. Any help is appreciated.

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2  
What have you tried? –  Code-Guru Jul 19 '12 at 20:06
    
Let's focus on one question at a time, please. –  Arkamis Jul 19 '12 at 20:06
    
$\sin(1/x)$ isn't uniformly continuous on $(0,1)$... –  Cocopuffs Jul 19 '12 at 20:11
    
Hint: a function that is uniformly continuous on $(0,1)$ must be bounded. –  David Mitra Jul 19 '12 at 20:11

1 Answer 1

up vote 1 down vote accepted

A continuous function defined on a bounded open interval is uniformly continuous iff both end point limit exist finitely. here your open interval is $(0,1)$ and $lim_{x\rightarrow 0}\frac{1}{\sin x}$ does not exist finitely so this is not uniformly continuous on $(0,1)$

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