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Is there any function $f:[\alpha,\beta]\rightarrow\mathbb{R}$ that is bound but not Henstock-kurzweil integrable? I assume such a function would have to be horrendously discontinuous but I am unable to construct one.

Edit: Perhaps we can defined a function on $[0,1]$ that equals $1$ on the Cantor set and $0$ otherwise. I might be wrong, but it seems like perhaps something like that would work.

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  • $\begingroup$ What does "dense graph" mean? $\endgroup$
    – David
    Commented Oct 31, 2018 at 7:55

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Yes. Notice Lebesgue and H-K integrals are equivalent for $f$ bounded with compact support.

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