= Solution
The <total variation seminorm on a domain> is
$$
\boxed{|Df|(\Omega)=\sup\left\{\int_\Omega f\,\operatorname{div}\zeta\,dx:\zeta\in C_c^1(\Omega;\mathbb R^2),\ \sup_x|\zeta(x)|_2\le1\right\}.}
$$
The class is symmetric under negation, so no additional absolute value is needed around the integral. The <bounded-variation space> is
$$
BV(\Omega)=\{f\in L^1(\Omega):|Df|(\Omega)<\infty\},\qquad
\|f\|_{BV}=\|f\|_1+|Df|(\Omega).
$$
Equivalently, 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.
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