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Thread: Please help me solve this problem. Thank you

  1. #1
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    Please help me solve this problem. Thank you

    Let f(x)={0 x rational ; 1 x irrational} Let P be any partition of [0,1]. Compute L(f,P) and U(f,P)/ Compute L(f) and U(f). Is f integrable on [0,1]
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  2. #2
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    Re: Please help me solve this problem. Thank you

    consider any interval $I$ of width $\delta$ that's a subset of $[0,1]$

    does $I$ contain a rational number?

    If so what will $L(f,I)$ be?

    does $I$ contain an irrational number?

    If so what will $U(f,I)$ be?

    now consider a partition of $[0,1]$ which will be made up of intervals such as $I$ above

    given what you know about $L,U$ of $I$ what is $L(f,P),~U(f,P)$ ?

    If $f$ is integrable on $[0,1]$ what is required of $L(f,P)$ and $U(f,P)$ ?

    Does that hold true in this case?

    So is $f$ integrable on $[0,1]$ ?
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