Are bound functions always Henstock-kurzweil integrable?

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.

• What does "dense graph" mean? – David Oct 31 '18 at 7:55

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