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Math Help - Trouble with Uniform continuity calculation

  1. #1
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    Trouble with Uniform continuity calculation

    So i have to prove that f(x)=\frac{1}{\sqrt{x}} is uniformly continuous using the definition of uniform continuity...

    aka I have to start with |\frac{\frac{1}{\sqrt{x}}-\frac{1}{\sqrt{c}}}{x-c} + \frac{1}{2\sqrt{c^3}}|

    and knowing that |x-c|<\delta, I have to make the first inequality less than \epsilon.

    I am getting so lost in the calculation and this problem is due tomorrow, someone please help....
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  2. #2
    Senior Member Pinkk's Avatar
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    Uniformly continuous on what domain?
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  3. #3
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    x>0
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  4. #4
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    Does anyone have any idea??
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  5. #5
    Senior Member Pinkk's Avatar
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    Well, it is NOT uniformly continuous on (0,\infty), and it's easy to show this as follows: Consider the sequence s_{n} = \frac{1}{n^{2}}, which is Cauchy, but f(s_{n}) = n is not Cauchy, and so f cannot be uniformly continuous. Not exactly sure how to show this using the \epsilon - \delta definition though.
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  6. #6
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    yea sorry the problem says for x>0
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  7. #7
    Senior Member Pinkk's Avatar
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    Yes, and for that interval it is NOT uniformly continuous and to somehow use the epsilon-delta definition, you have to show that \forall \, \delta > 0, \exists \, \epsilon > 0 such that for some x,y \in (0,\infty ), |x-y| < \delta and |f(x) - f(y)| \ge \epsilon.
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