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Some new results on the continuity and differentiability of functions of several real variables. (English) Zbl 1076.26009

This book includes many results which cannot be found in the usual textbooks and monographs devoted to the analysis of real functions of several variables. A large part refers to results obtained by the author, who is a well-known specialist in the field. Chapter 1 is entitled “Separately partial continuity from different points of view and continuity” and includes results related to: the equivalence between continuity and separately strong partial continuity; the equivalence between continuity and separately angular partial continuity; the nonexistence of the limit and the continuity, continuity of functions of two variables; continuity in the wide and its application; the limit in the wide and its application; partial continuity with respect to one of the variables, uniformly with respect to the other variable; unilateral limit and continuity of functions of two variables. Chapter II “Separately partial differentiability in various senses and differentiability” includes results such as: equivalence between differentiability and the finiteness of an angular gradient; finiteness of a strong gradient implies differentiability. Chapter III “Twice differentiability, Bettazzi derivative and mixed partial derivatives” includes conditions of twice differentiability; properties of Bettazzi derivative of functions of two variables; sufficient conditions for equality of mixed partial derivatives of functions of two variables. Chapter IV “On double indefinite integral and absolutely continuous functions of two variables” deals with: differentiability of an indefinite double integral and of an absolutely continuous function of two variables; Lebesgue’s intense points and finiteness at these points of a strong gradient of an indefinite integral; repeated and mixed partial derivatives of an indefinite integral; twice differentiability of an indefinite integral.
Reviewer’s remark. The notion of a “Bettazzi derivative” studied in Chapter III, §2 and introduced by Bettazzi in 1884 was rediscovered 50 years later by Karl Bögel [“Mehrdimensionale Differentiation von Funktionen mehrerer Veränderlichen”, J. Reine Angew. Math. 170, 197–217 (1934; Zbl 0008.25005/JFM 60.0215.02)] apparently ignoring Bettazzi’s work. Unfortunately, most authors refer to Bögel when dealing with the Bettazzi derivative; so, Dzagnidze’s results should be checked with the literature devoted to Bögel’s work (see also [K. Bögel, J. Reine Angew. Math. 173, 5–30 (1935; Zbl 0011.05903/JFM 61.0254.02)].

MSC:

26B05 Continuity and differentiation questions
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
26B15 Integration of real functions of several variables: length, area, volume
26-02 Research exposition (monographs, survey articles) pertaining to real functions
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