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Math Help - f is not integrable over [0,6]

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    f is not integrable over [0,6]

    define  f:[0,6] \to \mathbb{R}
    f(x)=\left\{\begin{array}{ll}x&\;\;if\,\,x \ is \ rational\\0&\;\; \ x \ is \ irrational \end{array}\right.

    prove that f is not integrable over [0,6]
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  2. #2
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    by contradiction .....
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by flower3 View Post
    define  f:[0,6] \to \mathbb{R}
    f(x)=\left\{\begin{array}{ll}x&\;\;if\,\,x \ is \ rational\\0&\;\; \ x \ is \ irrational \end{array}\right.

    prove that f is not integrable over [0,6]
    Exact same argument as you have seen before. It suffices to prove that isn't integrable on [0,1] then given any partition \left\{x_0,\cdots,x_n\right\} then \sup_{x\in[x_{j-1},x_j]}f(x)=x_j>0,\inf_{x\in[x_{j-1},x_j]}f(x)=0
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