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Math Help - Riemann Integrability

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    Riemann Integrability

    Is this riemann integrable on [0,1]?

    f(x)= 0 when x is rational and f(x)= 1 when x is irrational
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by CrazyCat87 View Post
    Is this riemann integrable on [0,1]?

    f(x)= 0 when x is rational and f(x)= 1 when x is irrational
    Is the set of discontinuities of measure zero? What's \sup_{x\in I}f(x) and \inf_{x\in I}f(x) for ANY subinterval I\subseteq [0,1]?
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    I guess I'm having trouble with this material since I'm unsure of what you're asking here....
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by CrazyCat87 View Post
    I guess I'm having trouble with this material since I'm unsure of what you're asking here....
    What is the biggest and smallest values f takes on any subinterval?
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    I understand that it's at most 1 and 0, I just don't know how to prove this using the definition, which I think is what I'm supposed to do
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by CrazyCat87 View Post
    I understand that it's at most 1 and 0, I just don't know how to prove this using the definition, which I think is what I'm supposed to do
    The point is this. If you give me any partition of [0,1] into subintervals, let's just call them I_1,\cdots,I_n, we find that U(P,f)=1. Why? Well, let's look at something. These intervals are presumably non-degenerate (not one point) and so in any of these intervals there are both irrational and rational numbers. So, \sup_{x\in I_k}f(x)=1 for any subinterval since it must contain an irrational. So, U(P,f)=\sum_{j=0}^{n}\sup_{x\in I_j}f(x)\Delta I_j=\sum_{j=1}^{n}\delta I_j=1-0=1. But, why does this last sum equal one? The way I wrote it should give you a clue. Start with that .
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