 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 $\mathrm{TV}(u)=2$. Directly from the test-function definition, $\int u{\phi }^{\prime }=\phi (b)-\phi (a)\le 2$, and two disjoint normalized smooth bump functions attain the bound. Values at the endpoints themselves do not change the $L^1$ function.