On Schwarz differentiability, III

Abstract

h~0 lower Schwarz derivative or symmetric derivative for the function f(x) at the point x and denoted byf(')(x) and fv)(x), respectively. If, however, f(')(x) and f(') (x) are finite and equal, then the common value is called the Schwarz derivative [14] or the symmetric derivative [6] off(x) at x and is denoted by f(')(x). It is clear that if the ordinary derivative f'(x) exists at the point x, then f(')(x) also exists and f(')(x)=f'(x). converse, however, is not true. It is natural to ask whether the main properties of the ordinary derivative are also possessed by Schwarz derivative: Indeed, it has been shown [14] that if F(x) is continuous in [a, b] and the second Schwarz derivative F()(x) = 0 everywhere on [a, b], then F(x) is a linear function. Some sufficient conditions in terms of Schwarz derivative have also been obtained [14] in order that a function may be convex downwards. In an earlier paper [12] the author has obtained, under some conditions, Rolle's Theorem and Mean Value Theorem for Schwarz derivative. Some other results on Schwarz derivative can also be found in most of the cited references. It is well known that if f '(x) exists in [a, b] then it possesses Darboux's property, viz. if f'(c) fl) the set E(~, fl)=E{x:x~[a, b]; ~ >f'(x)>fl} is either void or of positive measure. This property of the derived function has been referred to by LAHIRI [7] as Clarkson's Property, who has extended Clarkson's theorem by showing that the set E(~, fl) if it is non-void, is metrically dense in itself. Recently HSIANG [5] has obtained the following property of the set E(~, fl): The set E(~, fl) gives rise to a set of non-overlapping and non-abutting open subintervals {Ii} in the space [a, b] and a closed set F which is the complementary set of {Ii} with respect to [a, b] such that E(~, fl) is void in each Ii and metrically dense everywhere in F.