What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative?

More precisely, I'd like to see an example (and a plot using Mathematica or Matlab) of

- a function $$u_1 \in BV(\mathbb R^2; \mathbb R)$$ with only jump part in the derivative $$Du_1 = D^{jump} u_1$$
- and of a function with only Cantor part in the derivative: $$u_2 \in BV(\mathbb R^2; \mathbb R)$$ with $$Du_2 = D^{cantor} u_2$$

A related more general question is Heuristic and graphic representation of BV functions and their singularities

Clearly one could take a one dimensional example $f \in BV(\mathbb R)$ and then consider $g(x_1,x_2) := f(x_1)$. However, I'd like to see a "genuinely" two-dimensional example (if it exists).