Total variation of an interval indicator
= Total variation of an interval indicator
For $a<b$ strictly inside a one-dimensional domain, the indicator $u=1_{[a,b]}$ has <distributional derivative> $Du=\delta_a-\delta_b$, hence $\operatorname{TV}(u)=2$. Directly from the test-function definition, $\int u\varphi'=\varphi(b)-\varphi(a)\leq2$, and two disjoint normalized <smooth bump functions> attain the bound. Values at the endpoints themselves do not change the $L^1$ function.