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Math Help - Real analysis

  1. #1
    MHF Contributor Amer's Avatar
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    May 2009
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    Jordan
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    Real analysis

    Hi there

    show that f(x) is continuous at R




    where 0 < a < 1 & ab>1+3/2T ......... T : means : Bie

    can you just tell me how I can prove it , what theorems or definitions I should use to prove it ?


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  2. #2
    Member
    Joined
    Nov 2006
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    Florida
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    First let f_n(x)=a^n\cos(b^n\pi x), then |f_n(x)|\le a^n=M_n. Well clearly each f_n(x) is continuous. Now define f(x)=\sum_{n=0}^\infty f_n(x), so if \sum f_n\to f uniformly we will know that f is continuous.

    So by the M-test we have that \sum f_n\to f uniformly since \sum_{n=0}^\infty M_n<\infty (geometric series with common ratio less than 1). If you don't know what the M-test is then just follow the above link.

    As for the condition on ab it is not needed to show continuity, but if I am correct; I think that the next step would be to show that the function is nowhere differentiable. This is where the condition is most likely used.
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