## 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

Zbl 0918.26003