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.


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


Zbl 0918.26003