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A unified theory of bilateral derivates. (English) Zbl 1016.26008

Continuing his work on differentiation [K. M. Garg, “Theory of differentiation. A unified theory of differentiation via new derivate theorems and new derivatives” (1998; Zbl 0918.26003)], the author presents a unified theory of bilateral derivates (= biderivates), by means of two fundamental theorems in terms of bimonotonicity and bi-Lipschitz properties, from which two more theorems are obtained, dealing with the properties of a function on a portion of a given set, and one more biderivate theorem on the Baire class of biderivates. The median \(Mf\) of \(f: X\to\mathbb{R}\), where \(X\subset\mathbb{R}\), is defined as the multifunction \(Mf(x)= [\underline D f(x),\overline Df(x)]\), where \(x\in X'=\) the set of all limit points of \(X\) in \(X\). By means of the theorems on biderivates some theorems on the median are obtained, leading to a unified approach to differentiation including, for instance, the Goldowski-Tonelli theorem, various mean-value theorems and the Darboux property of median and derivative, the Denjoy property of derivatives, results on the Baire class of derivatives and medians, a biderivate version of the classical Denjoy-Young-Saks theorem as well as some more theorems due to Denjoy, Young, Choquet, Zahorski, Kronrod, and Marcus. By its elegance, this article has also aesthetic qualities.

MSC:

26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26A30 Singular functions, Cantor functions, functions with other special properties

Citations:

Zbl 0918.26003
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