Differentiability and integrability in \(n\) dimensions with respect to \(\alpha\)-regular intervals. (English) Zbl 0764.28005

The regularity of an interval (i.e., the ratio of its shortest and its longest edge) plays an important role in multidimensional differentiation and integration theory. The present paper is devoted to an analysis of its relevance in multidimensional generalized Riemann integrals and in differentiability theory for additive interval functions. If, for \(\alpha<1\), a concept of \(\alpha\)-regular differentiability of an additive interval function is defined in a natural way, it is proved in the paper that the \(\alpha\)-differentiability at one point implies the \(\beta\)-differentiability at this point for any \(0<\beta<\alpha\). The situation is quite different for \(\alpha\)-regular integrability based upon the Kurzweil-Henstock Riemann-type definition restricted of \(\alpha\)-regular intervals. In this case, it is shown that given \(\alpha\in]0,1[\), there exists a function \(f=f_ \alpha\) which is \(\alpha_ 1\)-regularly integrable for every \(\alpha_ 1\in]\alpha,1[\) and is not \(\alpha_ 2\)-regularly integrable for every \(\alpha_ 2\in]0,\alpha[\) on the interval \([-1,2]^ n\).


28A15 Abstract differentiation theory, differentiation of set functions
26B99 Functions of several variables
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
26A39 Denjoy and Perron integrals, other special integrals
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