On the mixed derivatives of a separately twice differentiable function (Q1704511)

The main result of the paper is the following Theorem. Let \(f:[0,1]^2\rightarrow\mathbb{R}\) be a separately twice differentiable function and \(f_{xx}^{''},f_{yy}^{''}\in L^2([0,1]^2)\). Then: 1) mixed derivatives \(f_{xy}^{''}\) and \(f_{yx}^{''}\), which are equal and \(f_{xy}^{''},f_{yx}^{''}\in L^2([0,1]^2)\), exist almost everywhere; 2) a set \(A\subset [0,1]\) with \(\mu(A)=1\) exists such that \(f_{x}^{'}\) is continuous with respect to \(y\) at every point of the set \(A\times [0,1]\); 3) \(f\in C([0,1]^2)\). Note that the condition of square integrability of \(f_{xx}^{''}\) and \(f_{yy}^{''}\) is essential, namely, in the [\textit{V. Mykhaylyuk} and \textit{A. Plichko}, Colloq. Math. 141, No. 2, 175--181 (2015; Zbl 1339.26029)] the authors constructed a separately twice differentiable function \(f:[0,1]^2\rightarrow\mathbb{R}\) for which the mixed partial derivative \(f_{xy}^{''}\) does not exist on a set of positive measure.