A Unified Theory of Bilateral Derivates

Abstract

We present here a unified theory of bilateral derivates, which we call here briefly "biderivates". This unified theory is achieved with the help of two new fundamental theorems on biderivates, called the First and Second Biderivate Theorems. These two biderivate theorems are obtained in terms of bimonotonicity and bi-Lipschitz properties of a function on a set which also depend on the values of the function outside the set like the properties $VB_*$ and $AC_*.$ From these two biderivate theorems we further deduce the Third and Fourth Biderivate Theorems, which deal with the properties of a function on a portion of a given set and the Fifth Biderivate Theorem on the Baire class of biderivates. Next, given $f:X\to\mathbb{R},$ where $X\subset \mathbb{R},$ let $X^\prime$ denote the set of limit points of $X$ in $X.$ Then we define the ``median'' $Mf$ of $f$ to be the multifunction $Mf(x)= [\underline Df(x),\overline Df(x)],$ $x\in X^\prime.$ We deduce here from the above biderivate theorems five basic median theorems which deal with the properties of the median. The various known results on biderivates are deduced here from these biderivate and median theorems many of which are strengthened in this process. As the known results on biderivates were obtained earlier by ad~hoc methods, the present theory provides a synthesis of these results and brings out their inter-relations leading thereby to a unified theory. In particular, we deduce from one of the median theorems several monotonicity theorems in terms of biderivates including an extended version of the well known Goldowski-Tonelli theorem. Further, we deduce from another median theorem a mean-value theorem in terms of median and derivative, and the Darboux property of median and derivative. Also, the Denjoy property of derivatives is obtained from the Third Biderivate Theorem and results on the Baire class of derivatives and medians are obtained from the Fifth Biderivate Theorem. Also, a biderivate version of the classical Denjoy-Young-Saks theorem is obtained from the First Biderivate Theorem. From the above mentioned biderivate and median theorems we also deduce some other known theorems of Denjoy, Young, Choquet, Zahorski, Kronrod, Fort and Marcus, along with the following two new results: (i)a theorem on biderivates similar to a fundamental theorem of Denjoy on unilateral derivates; and (ii)a theorem on median derivates similar to a theorem of Morse on unilateral derivates.