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On the differentiability of functions of two variables and of indefinite double integrals. (English) Zbl 0836.26007
Kokilashvili, V. (ed.), Collected papers in function theory. Dedicated to the memory of Georgian mathematician Academician Boris Khvedelidze. Tbilisi: Publishing House GCI, Proc. A. Razmadze Math. Inst. 106, 7-48 (1993).
“The first part of this paper deals with the necessary and sufficient conditions for a function of two variables to be continuous or differentiable. In the second part it is shown that the strong gradient of \(F(x,y)\) is finite, in particular, that the Lebesgue double integral \(F(x,y)\) is differentiable in the sense of Stolz. The continuity and differentiability of the partial derivatives \(F_x'\) and \(F_y'\) are investigated. Finally, the notion of a Lebesgue point in the strong sense is introduced and the corresponding results are proved”. (Author’s abstract).
In his approach, the author introduces some interesting notions such as: separately angular continuity, separately strong continuity and angular partial derivative.
Remarks of the reviewer. Some related results were obtained by the reviewer [Dokl. Akad. Nauk SSSR 112, 812-814 (1957; Zbl 0085.04402); C. R. Acad. Sci., Paris 246, 522-524 (1958; Zbl 0078.04603); see also Proc. Am. Math. Soc. 12, 562-564 (1961; Zbl 0100.05404)].
For the entire collection see [Zbl 0798.00019].

26B05 Continuity and differentiation questions
26B15 Integration of real functions of several variables: length, area, volume