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    Riemann integrable function on [0, 1]

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    Consider a function fâ?¶[0,1]â?'[0,1] given by
    f(x)={(1/q, if x=p/q,where p,qâ??N are coprime,
    0, if x is irrational,
    1, if x=0.

    (a) Show that if xâ??[0,1] is rational, then f is not continuous at x.

    (b) Show that L(f,P)=0 for any partition P of [0,1].

    (c) Show that for any qâ??N the number of elements in the set
    X(q)={xâ??[0,1]:f(x)â?¥1/q }
    can be bounded by q(q+3)�2. (Better bounds are possible, but not required.)

    (d) Using that X(q) is finite, show that f is continuous at any irrational xâ??[0,1].

    (e)Show that for any ?>0, we can find a partition P_? of [0,1] such that U(f,P_?)<?.

    (f)Deduce that f is Riemann integrable and
    â?«^1_0 f(x)dx=0.

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    https://brainmass.com/math/graphs-and-functions/riemann-integrable-function-on-0-1-395210

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