Is it fair to say that [Alberti rank one theorem][1] means that a BV functions $u \in BV(\mathbb{R}^2)$ has $Du = D^{cantor}u$ if and only if it has a **jump discontinuity across a curve** that is not smooth (as it would be in the case $Du = D^{jump} u$) but **of "fractal type"** (for example **Koch** curve)? Or maybe many jump discontinuities across Koch-type curves oriented in the same direction?

If this is the case, how can one make this intuition rigorous?

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Related questions are asked in https://mathoverflow.net/questions/328493/heuristic-and-graphic-representation-of-bv-functions-and-their-singularities  and https://mathoverflow.net/questions/328779/concrete-example-of-bv-function-u-mathbbr2-to-mathbbr-with-singular-de?noredirect=1&lq=1.


  [1]: https://mathoverflow.net/questions/315401/meaning-of-alberti-rank-one-theorem