On the measurability of extreme partial \(\mathcal {I}\)-approximate derivatives (Q1890599)

For a function \(F: R\to R\) (\(R\) -- the real line) the notion of the \(I\)- approximate upper right (lower right, upper left, lower left, bilateral upper and bilateral lower) derivative at \(x_ 0\) is defined as the corresponding extreme limit of \((F(x)- F(x_ 0))/(x- x_ 0)\) as \(x\) tends to \(x_ 0\) in \textit{E. Lazarow} and \textit{W. Wilczyński} [Rad. Mat. 5, No. 1, 15-27 (1989; Zbl 0685.26004)]. For a function \(F: R^ 2\to R\) having the Baire property in the direction of the \(x\)-axis the \(I\)-approximate upper right partial derivative of \(F\) at \((x_ 0, y_ 0)\) in the \(x\) direction is defined as a corresponding extreme limit of \((F(x, y_ 0)- F(x_ 0, y_ 0))/(x- x_ 0)\). The other extreme \(I\)- approximate partial derivatives in the \(x\) direction are defined similarly. In a similar way can be defined the extreme \(I\)-approximate partial derivatives in the direction of the \(y\)-axis. The authors prove that if a function \(F: R^ 2\to R\) has the Baire property then the extreme \(I\)-approximate partial derivatives also have the Baire property.