Bounded-variation step outside W11 (${\chi }_{\{x_1>1/2\}}\in BV((0,1{)}^2)\setminus W^{1,1}((0,1{)}^2)$)

The distributional derivative of this indicator function is $e_1{\mathcal{H}}^1$ on the vertical segment $x_1=1/2$, so its total variation seminorm on a domain is one. This is singular with respect to planar Lebesgue measure, whereas Sobolev space weak derivatives in $W^{1,1}$ have integrable densities. Its bounded-variation space norm is $1/2+1=3/2$.