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Math Help - Continuity

  1. #1
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    Continuity

    Determine the values where the function is continuous.

    f(x)={x for rational numbers
    {1-x for irrational numbers

    I drew a sketch of what the graph would look like, and it seems that the function is not continuous anywhere. The two "lines" cross at (.5, .5), but .5 is rational, not irrational, so it doesn't satisfy the second half of the function. Is this logic correct?
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  2. #2
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    Re: Continuity

    Quote Originally Posted by lovesmath View Post
    Determine the values where the function is continuous.
    f(x)={x for rational numbers
    {1-x for irrational numbers
    Suppose that \alpha>0 is irrational and \beta=\frac{\alpha}{2}.
    Then \beta is irrational.
    Thus f(.5+\beta)=.5-\beta~\&~f(.5-\beta)=.5+\beta.
    Now if |.5-x|<\beta what about |f(.5)-f(x)|~?
    Last edited by Plato; November 3rd 2012 at 08:00 AM.
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  3. #3
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    Re: Continuity

    f(x) is continuous at x = 1/2 if lim f(x) as x→1/2 =1/2 and f(1/2) = 1/2.
    So show that, given ε, δ exixts st│f(x)-1/2│ < ε if │x-1/2│ < δ

    x rational: │f(x)-1/2│= │x-1/2│ so δ=ε
    x irrational: │f(x)-1/2│= │1-x-1/2│=│x-1/2│so δ=ε

    So │f(x)-1/2│ < ε if │x-1/2│< ε and lim exists. Also, f(1/2) = 1/2
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