Some observations on regular dependence of total variation on parameters (Q1772926)

Let \(I\) be an interval, \(X\) the set of parameters, and \(f:X\times I\to\mathbb{R}\) a function. Denote by \(v(x)\) the total variation of the section \(f_{x}\) on \(I\), \(x\in X\). The authors study the regular dependence (measurability, Baire property, quasi-continuity) of \(v\) on the parameter. They give sufficient conditions for the regularity of \(v\), and construct some interesting counterexamples. The main role is played by different assumptions on the regularity of sections \(f^{t}\), \(t\in I\). The paper is concluded with 4 open problems. The authors continue the investigations initiated by \textit{M. Balcerzak, A. Kucia} and \textit{A. Nowak} [Real. Anal. Exch. 29, No. 2, 921--930 (2003-2004; Zbl 1070.26009)].