change of sign of $\xi$, this characterizes the triplet $(a,b,\xi)$ which is then denoted by $(u^+,u^-,\nu_u)$. The set of approximate jump points is denoted by $J_u$. The following theorem holds [@Am_Fu_Pal].

\[teo:federer\] The set $S_u$ is countably $\H^{d-1}$-rectifiable and $\H^{d-1}(S_u\setminus J_u)=0$. Moreover $Du{ \!\!\mbox{{ \Large$\llcorner$}} }J_u=(u^+-u^-)\nu_u\H^{d-1}{ \!\!\mbox{{ \Large$\llcorner$}} }J_u$ and $$Tan^{d-1}(J_u,x)=\nu_u(x)^\perp$$ for $\H^{d-1}$-a.e. $x\in J_u$.

We write the Radon-Nikodym decomposition of $Du$ as $Du= {\nabla}u \dx+D^s u$. Setting $D^c u=D^s u { \!\!\mbox{{ \Large$\llcorner$}} }({\Omega}\setminus S_u)$ we get the decomposition $$Du={\nabla}u \dx+ (u^+-u^-)\nu_u\H^{d-1}{ \!\!\mbox{{ \Large$\llcorner$}} }J_u+D^c u.$$ Moreover the Cantor part is such that if $\H^{d-1}(E)<\infty$, we have $|D^cu|(E)=0$ [@Am_Fu_Pal Thm. 3.92]. In particular, we have the following useful consequence $$\label{BVproperty}
\H^{d-1}(E)=0\quad\Longrightarrow\quad |Du|(E)=0.$$ We frequently use the notation $[u]$ for the jump function $(u^+-u^-):J_u{\rightarrow}{\mathbf{R}}$.\
When $d=1$ we use the symbol $u'$ in place of ${\nabla}u$ and $u(x^\pm)$ to indicate the right and left limits of $u$ at $x$.

Let us introduce the space of special functions of bounded variation and a variant: $$SBV({\Omega}):=\{u\in BV({\Omega}):D^cu=0\},$$ $$GSBV({\Omega}):=\{u\in L^1({\Omega}):\max(-T,\min(u,T))\in SBV({\Omega})\;\forall T\in{\mathbf{R}}\}.$$ Eventually, in Section \[sec:LocRes\], the following space of piecewise constant functions will be useful. $$\P({\Omega})=\{u\in GSBV({\Omega}):{\nabla}u=0\}.$$ To conclude this section we recall the slicing method for functions with bounded variation. Let $\xi\in\S^{d-1}$ 