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Math Help - Dirichlet function

  1. #1
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    Dirichlet function

    I am trying to prove that f(x)=xD(x) is only continuous at x=0, however I don't know if I can prove it with assuming two different cases for continuity, in the neighbourhood of x=a when x is rational and irrational.. Not sure if it's possible to assume this, would appreciate if I can get feedback. Thank you.(In the picture I meant proving continuous , not convergence)
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  2. #2
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    Re: Dirichlet function

    Quote Originally Posted by davidciprut View Post
    I am trying to prove that f(x)=xD(x) is only continuous at x=0, however I don't know if I can prove it with assuming two different cases for continuity, in the neighborhood of x=a when x is rational and irrational.. Not sure if it's possible to assume this, would appreciate if I can get feedback.
    In every neighborhood there are both rational and irrational numbers. So of course you need cases.
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    Re: Dirichlet function

    So my proof is correct?
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  5. #5
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    Re: Dirichlet function

    Where is the problem? I know that D(x) is not continuous at any point but f(x)=xD(x) is continous at x=0.
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    Re: Dirichlet function

    Quote Originally Posted by davidciprut View Post
    Where is the problem? I know that D(x) is not continuous at any point but f(x)=xD(x) is continous at x=0.
    You are correct. The function
    is continuous at zero and nowhere else.

    If . That shows it is continuous at zero.

    If any neighborhood of x contains both rational and irrational numbers which means that f cannot be continuous at x.
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