Some observations on regular dependence of total variation on parameters

Abstract

Let $I$ be a nondegenerate interval and let $X\neq\emptyset$ be a set. For a function $f:X\times I \to {\mathbb{R}}$ and $x\in X$ define $v(x)$ as the total variation of the section $f_x$ on $I$. We investigate the regular dependence (measurability, Baire property, etc.) of $v$ on the regularity of the sections $f^t$.