The updated version can be found here.

Let $S\subset \mathbb R^N$ be a $N-1$ rectifiable set with $\mathcal H^{N-1}(S)<\infty$.

My question, for each $x_0\in S$, would it be possible to choose a ball (cube) $B$ centered at $x_0$ with radius $r>0$ so that:

I can choose a direction vector $\nu$ so that each slice $B_x$, for $x\in B_\nu$, only intersect with $S\cap B$ once?

Here by slice $B_x$ please refer to the following definition:

\begin{align} \begin{cases} \pi_\nu = \{x\in\mathbb R^N:\,<x,\nu>=0\};\\ B_x=\{t\in R:\, x+t\nu\in B\}\,\,(x\in\pi_\nu);\\ B_\nu = \{x\in\pi_\nu:\,B_x\neq \varnothing\}. \end{cases} \end{align}

Little background: Here my set $S$ is the jump set of a $BV$ function in $\mathbb R^N$.

Thank you!