# Plot BV functions with singular derivatives in multiple space dimension

In 1-dimensional setting, two key examples of $$BV$$ functions (of bounded variation) $$u: \mathbb R \to \mathbb R$$ are the Heaviside function, whose derivative is the Dirac delta concentrated at $$0$$. It is an example of $$BV$$ function such that $$Du = D^{jump}u$$ and the Cantor staircase function, whose derivative is the $$\log_3(2)$$-dimensional Hausdorff measure restricted to the Cantor set. It is an example of BV function such that $$Du = D^{cantor}u$$. It is straightforward to plot these functions using Mathematica.

However, I'd like to see the graph of two functions $$u:\mathbb R^2 \to \mathbb R^2$$ which are $$BV$$ and have respectively $$Du=D^{jump}u$$ and $$Du=D^{cantor}u$$ (that is, only jump or Cantor part respectively in the derivative).

How can I use Mathematica to plot such functions and get a representation of their derivatives?

One motivation for this is to see what $$BV$$ in multiple space dimensions look like at the jump set and to check the heuristic meaning of Alberti rank-one theorem on the "direction" of singularities. For more context and motivation, see a related question on MathOverflow.

• @HenrikSchumacher Yes, thank you for correction. – Riku Apr 21 at 14:59