A unified theory of bilateral derivates (Q1852446)

Continuing his work on differentiation [\textit{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.