vectors have length one. Repeating the proof from [@EG] we get that this definition coincides with the divergence definition. Then we extend it to the case of a one-dimensional compact connected manifold in the following way:

We say that $f \in BV(\partial\Omega)$, if after removing from $\partial\Omega$ a point $p$ of approximate continuity of $f$ we have $f \in BV(\partial\Omega \backslash \{ p \})$. The norm is defined to be

$$\| f \|_{BV(\partial\Omega)} = \| f \|_1 + \| f \|_{BV(\partial\Omega \backslash \{ p \})}.$$

This definition does not depend on the choice of $p$, as in dimension one the total variation on disjoint intervals is additive, thus for different points $p_1, p_2$ we get that

$$\| f \|_{BV(\partial\Omega \backslash \{ p_1 \})} = \| f \|_{BV((p_1,p_2))} + \| f \|_{BV((p_2,p_1))} = \| f \|_{BV(\partial\Omega \backslash \{ p_2 \})},$$

where $(p_1,p_2)$ is an oriented arc from $p_1$ to $p_2$. Thus all local properties of $BV(\partial\Omega)$ hold; we shall recall the most important one for our considerations:

\[stw:bvdim1\] Let $E \subset \partial\Omega$ be a set of finite perimeter, i.e. $\chi_E \in BV(\partial\Omega)$. Then, if we take its representative to be the set of points of density $1$, then $\partial E = 