On the convergence of the integrals of a truncated Henstock-Kurzweil integrable function (Q1978859)

The function \(f:[a,b] \to \mathbb R\) is Kurzweil-Henstock integrable with \(\int _a^bf=A\) if and only if f is measurable and there are increasing sequences of positive integers \(M_k, N_k \) and a decreasing sequence of gauges \(\delta _k\) such that for all \(k\) and \(\delta _k\)-fine tagged partition \(P\) we have \(|S(f,P) - S(f_k,P)|<1/k\) and \(|S(f_k,P) - A|<1/k\), where \(S\) stands for the Riemann sum and \(f_k(x) = f(x)\) if \(-N_k \leq f(x) \leq M_k\), \(f_k(x) = M_k\) if \(f(x) \geq M_k\) and \(f_k(x) = -N_k\) if \(f(x) \leq -N_k\).