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Sequences and Series of Functions Question (Rudin Chapter 7)

  1. May 19, 2013 #1
    1. The problem statement, all variables and given/known data

    The problem is Exercise 8 from Chapter 7 of Rudin. It can be seen here:

    http://grab.by/mGxY


    2. Relevant equations



    3. The attempt at a solution

    It seems quite obvious to see that because [itex]\sum\left|c_n\right|[/itex] converges, [itex]f(x)[/itex] will converge uniformly.

    However, I am having a difficult time understanding why [itex]f(x)[/itex] will be continuous at all [itex]x\neq{x_n}[/itex]

    Any help with understanding this second part of the proof would be greatly appreciated. Thanks!
     
  2. jcsd
  3. May 19, 2013 #2
    Is this as simple as stating that there is a jump discontinuity at [itex]x=x_n[/itex], because the left-hand limit = 0, and the right-hand limit = 1?
     
  4. May 19, 2013 #3

    verty

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    Which c_n's does I(x - x_n) choose to exclude from the sum?
     
  5. May 19, 2013 #4
    I'm not quite sure I understand your question. Do you mean for me to say that [itex]c_n[/itex] is excluded from the sum when [itex]I(x-x_n)≤0[/itex]?
     
  6. May 19, 2013 #5

    verty

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    I can't really say more without being too helpful.
     
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