[proofplan] The proof combines three ingredients: the Radon--Nikodym decomposition splits $Du$ into absolutely continuous and singular parts; Besicovitch's differentiation theorem and blow-up analysis identify the jump set from the singular part and show that the blow-up at jump points converges to half-space indicators; and the Preiss--Mattila rectifiability criterion gives that the jump set is countably $\mathcal{H}^{n-1}$-rectifiable. The Cantor part $D^c u$ is what remains after extracting the jump part from $D^s u$. [/proofplan] [step:Decompose $Du$ into absolutely continuous and singular parts via Radon--Nikodym] By the Radon--Nikodym theorem applied to the vector measure $Du$ and Lebesgue measure $\mathcal{L}^n$: \begin{align*} Du = D^a u + D^s u, \end{align*} where $D^a u = \nabla u \, d\mathcal{L}^n$ (with $\nabla u \in L^1(U; \mathbb{R}^n)$ the approximate gradient) and $D^s u \perp \mathcal{L}^n$. [/step] [step:Split the singular part into jump and Cantor components via blow-up analysis] The Besicovitch--Federer differentiation theorem shows that at $\mathcal{H}^{n-1}$-a.e. $x_0 \in J_u$ (the jump set), the rescaled measures $r^{1-n}D\mathbb{1}_E(x_0 + r\,\cdot)$ converge to a flat $(n-1)$-dimensional measure, and the Radon--Nikodym derivative of $|D^s u|$ with respect to $\mathcal{H}^{n-1} \lfloor J_u$ equals $|u^+ - u^-|$. Define: \begin{align*} D^j u := (u^+ - u^-)\,\nu_u \, \mathcal{H}^{n-1} \lfloor J_u, \qquad D^c u := D^s u - D^j u. \end{align*} The remaining measure $D^c u$ is singular with respect to both $\mathcal{L}^n$ and $\mathcal{H}^{n-1} \lfloor J_u$. The blow-up analysis shows that at $|D^c u|$-a.e. point, the measure has a diffuse tangent rather than codimension-one concentration, so $|D^c u|(A) = 0$ for every set $A$ with $\mathcal{H}^{n-1}(A) < \infty$. [/step] [step:Establish rectifiability of $J_u$ via the Preiss--Mattila criterion] At each point $x \in J_u$, the blow-up $r^{1-n}Du(x + r\,\cdot)$ converges as $r \to 0$ to $(u^+(x) - u^-(x))\,\delta_{\{y \cdot \nu_u(x) = 0\}}$, a codimension-one flat measure. The Preiss--Mattila rectifiability theorem (a Radon measure with $(n-1)$-dimensional tangent measures $\mathcal{H}^{n-1}$-a.e. is supported on a countably $\mathcal{H}^{n-1}$-rectifiable set) gives the rectifiability of $J_u$: there exist countably many $C^1$ hypersurfaces $\{S_k\}$ with $\mathcal{H}^{n-1}(J_u \setminus \bigcup_k S_k) = 0$. [/step] [step:State the full decomposition and mutual singularity] Assembling: \begin{align*} Du = \nabla u \, d\mathcal{L}^n + (u^+ - u^-)\,\nu_u \, \mathcal{H}^{n-1} \lfloor J_u + D^c u. \end{align*} The three components are mutually singular: $D^a u \ll \mathcal{L}^n$ while $D^j u + D^c u \perp \mathcal{L}^n$; and $D^j u \ll \mathcal{H}^{n-1} \lfloor J_u$ while $D^c u \perp \mathcal{H}^{n-1} \lfloor J_u$. [/step]