A necessary and sufficient condition for gauge integrability (Q1322638)

The authors showed that a function \(f\) is Henstock-Kurzweil integrable on \([a,b]\) if and only if there are a function \(F\) on \([a,b]\) and a strictly increasing differentiable function \(\phi\) mapping \([\alpha,\beta]\) onto \([a,b]\) such that \((F\circ \phi)'= (f\circ\phi)\cdot \phi'\) on \([\alpha,\beta]\). This condition has been studied by \textit{G. P. Tolstov} [Mat. Sb., n. Ser. 53(95), 387-392 (1961; Zbl 0109.039)]. For Henstock- Kurzweil integral, see, for example, the reviewer [Lanzhou lectures on Henstock integration (1989; Zbl 0699.26004)].