Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function.

Is the box counting dimension of the graph of $u$ equal to $N$? How can we prove it?

The analogous question for the Hausdorff dimension was asked in Hausdorff dimension of the graph of a BV function (where related issues are also mentioned).