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Math Help - Proving Continuity of a function

  1. #1
    Super Member redsoxfan325's Avatar
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    Proving Continuity of a function

    f(x)={0 if x is irrational | 1/n if x = m/n ∈ Q}

    I need to prove that f is continuous at every irrational point and that f has a simple discontinuity at every rational point.

    I have no idea where to begin because for all irrational x, there will always be a rational number in an arbitrarily small neighborhood around it.

    Please help me. Any input is appreciated.

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  2. #2
    MHF Contributor

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    You need to show that \lim_{x\rightarrow a} f(x)= 0 for all a. Yes, inside any interval around an irrational (or rational) number there exist a rational number. But, since there are only a finite number of m such that m/n is in such an interval, as the interval gets smaller, n must get larger and larger.
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  3. #3
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by HallsofIvy View Post
    You need to show that \lim_{x\rightarrow a} f(x)= 0 for all a. Yes, inside any interval around an irrational (or rational) number there exist a rational number. But, since there are only a finite number of m such that m/n is in such an interval, as the interval gets smaller, n must get larger and larger.
    How do I know that (the bolded part)? Because, I agree, if I can prove n gets larger and larger, then I'll be able to prove continuity. My worry was that the irrational point would contain some simple fraction like 1/2 in its neighborhood. How can I prove that n \rightarrow \infty as \frac{m}{n} \rightarrow x?
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