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Disprove uniform continuety

  1. Sep 11, 2011 #1
    i need to prove that [TEX]\frac{1}{\sqrt{x}}[/TEX] is not uniformly continues in (0,1)

    for epsilon=0.5

    [TEX]|\frac{1}{\sqrt{x}}-\frac{1}{\sqrt{y}}|=|[/TEX][TEX]]\frac{\sqrt{y}-\sqrt{x}}{\sqrt{xy}}\frac{\sqrt{y}+\sqrt{x}}{\sqrt {y}+\sqrt{x}}|[/TEX][TEX]=|\frac{y-x}{(\sqrt{y}-\sqrt{x})\sqrt{xy}}|[/TEX]

    i need to prove that the above exprseesion bigger then 0.5

    but i dont know what x and y to choose

  2. jcsd
  3. Sep 11, 2011 #2
    So, for each delta, you need to find an x and y such that |x-y| is smaller then delta but f(x)-f(y) is less than 1/2, as you have stated. But, realize two things:

    1) So long as |x-y| is smaller than delta, you can let |x-y| be anything you want (think about why this is true.) For example, if delta is bigger than zero, you can choose x and y such that (I'm going to write "d" for delta here) d/4 < |x-y| < d. In other words, you aren't proving that it is uniformly continuous, you are proving that it isn't (so think of the negation of the definition of uniform continuity.)

    2) multiply [TEX]\frac{1}{\sqrt{y}}-\frac{1}{\sqrt{x}}[/TEX] by [TEX]\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x}+\sqrt{y}}[/TEX] and then use the fact that |x-y| is bigger than something (see the above.)

    Does this make sense?
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