Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 340 / 5 / a / Solution

The total variation seminorm on a domain isThe class is symmetric under negation, so no additional absolute value is needed around the integral. The bounded-variation space isEquivalently, the distributional derivative is a finite vector-valued Radon measure, and the displayed quantity is its total mass in variation. For $f\in W^{1,1}(\Omega )$, it equals ${\int }_{\Omega }|\nabla f{|}_{2}dx$. Thus jumps are allowed, with their interface contributions measured by variation.