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Proving discontinuities

  1. Jul 5, 2009 #1
    1. The problem statement, all variables and given/known data
    Find (and prove as such) all discontinuities of the function [tex]g:[0,1]\to\mathbb{R}[/tex] given by
    [tex]g(x)=\sum_{n=1}^\infty \frac{1}{2^{2n-1}}\left\lfloor \frac{2^nx+1}{2} \right\rfloor[/tex]​
    where [tex]\lfloor\cdot\rfloor[/tex] is the greatest integer function.

    2. Relevant equations



    3. The attempt at a solution
    I'm pretty sure that the discontinuities all occur at [tex]x=(2k+1)2^{-m}[/tex] for positive integer [tex]k,m[/tex] since this is where the expression inside the greatest integer function is an integer. The thing is, I have no how to go about proving that these points are discontinuous. Can anyone steer me in the right direction?
     
  2. jcsd
  3. Jul 5, 2009 #2

    Office_Shredder

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    When x hits that point, note that every term in the sum is
    1) positive
    2) larger when you cross over the point

    And you get that as you approach (2k+1)2-m from the right, each term is strictly larger than when you approach from the left
     
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