Bounded-variation step outside W11
= Bounded-variation step outside W11
{title2=$\chi_{\{x_1>1/2\}}\in BV((0,1)^2)\setminus W^{1,1}((0,1)^2)$}
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 $1/2+1=3/2$.