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What would be a example of a function that is continuous, but not uniformly continous? will f(x)=1/x be a example? Give domain from (0,2), why? Strictly, in definitions

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up vote 13 down vote accepted

Clearly $\,\displaystyle{\frac{1}{x}}\,$ is continuous in $\,(0,2)\,$ as it is the quotient of two polynomials and the denominator doesn't vanish there.

Now, if the function was uniformly continuous there then

$$\forall\,\epsilon>0\,\,\exists\,\delta>0\,\,s.t.\,\,|x-y|<\delta\Longrightarrow \left|\frac{1}{x}-\frac{1}{y}\right|<\epsilon$$

But taking $\,\epsilon=1\,$ , then for any $\,\delta>0\,$ we take

$$x:=\min(\delta,1)\,\,,\,y=\frac{x}{2}\Longrightarrow |x-y|=\frac{x}{2}<\delta, \,\text{but nevertheless}$$

$$\left|\frac{1}{x}-\frac{1}{y}\right|=\left|\frac{1}{x}-\frac{2}{x}\right|=\left|\frac{1}{x}\right|\geq 1=\epsilon$$

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1  
+1 for this crystal way. – Babak S. Jan 30 '13 at 12:17

protected by Community Aug 14 '15 at 16:29

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