On regular \(J\)-differentiability (Q1289535)

The paper deals with so-called regular \({\mathcal I}\)-approximate differential of functions of two variables, where \({\mathcal I}\) is a sigma ideal of sets of the first category. The definition of this generalized differential differs from the total differential in such a way that a variable point \((x,y)\) tends to \((x_0,y_0)\) and belongs to some ``big'' union of the frontiers of squares centered at \((x_0,y_0)\). A typical result of the paper is the following theorem: Let \(f: \mathbb{R}^2\to \mathbb{R}\) be a function continuous with respect to \(x\) for \({\mathcal I}\)-almost every \(y\) and continuous with respect to \(y\) for \({\mathcal I}\)-almost every \(x\). Then \(f\) is regularly \({\mathcal I}\)-approximately differentiable \({\mathcal I}\)-a.e. if and only if \(f\) is partially differentiable with respect to \(x\) and \(y\) \({\mathcal I}\)-a.e.