Are bound functions always Henstock-kurzweil integrable?

Question

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.

Answer 1

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