Strongly balanced selections
The notion of path derivate and derivative was introduced by \textit{A. M. Bruckner}, \textit{R. J. O'Malley}, and \textit{B. S. Thompson} [Trans. Am. Math. Soc. 283, 97-125 (1984; Zbl 0541.26003)] and the notion of selective derivate and derivative by \textit{R. J. O'Malley} [Acta Math. Acad. Sci. Hung. 29, 77-97 (1977; Zbl 0345.26003)]. In this paper a new class of selections is defined, i.e., strongly balanced selections. It is proved that every selective derivative with respect to such selections can be realized as a path derivative. The main result is the following: If \(s\) is a strongly balanced selection and \(f\) is a monotone \(s\)- differentiable function then \(f\) is differentiable in the ordinary sense. (This is not so in general).
scientific article; zbMATH DE number 465107
scientific article