A counterexample for the change of variable formula in KH integrals (Q416417)

If \(F(x)\) and \(\psi(x)\) are Riemann integrals of the form \(\int^x_b f\,dx\) and \(\int^x_b \psi\,dx\), resp. then \(\psi\,f\circ\psi\), if defined, is Riemann integrable. Furthermore, the change of variable formula applies, giving NEWLINE\[NEWLINE\int^x_b \psi\,f\circ\psi\,dx= F(\psi(x))- F(\psi(b)).NEWLINE\]NEWLINE The author generalizes this theorem to the Kurzweil-Henstock (KH) integral; in other words, assuming that \(F\) and \(\psi\) are KH integrals of \(f\) and \(\psi\), resp. one would expect that \(\psi\,f\circ\psi\) be KH integrable.NEWLINENEWLINE The author proves that this is false, and produces a composition of two absolutely continuous functions that does not need to be ACG.