In this paper the following question is investigated. Given a natural number rr and numbers αj,βj for j = 0,1,…,r satisfying α0 < α1 < ⋯ < αr and ∑j=0rβjαkj={0r!if k=0,1,…,r−1if k=r , is there a 2π-periodic, r − 1 times continuously differentiable function ff such that lim suph↗0h−r(∑j=0rβjf(x+αjh))=lim suph↘0h−r(∑j=0rβjf(x+αjh))=∞, lim infh↗0h−r(∑j=0rβjf(x+αjh))=lim infh↘0h−r(∑j=0rβjf(x+αjh))=−∞ for every x ∈ ℝ? [ABSTRACT FROM AUTHOR]
In this paper we construct a differentiable function F : R → R that does not have a third order symmetric Riemann derivative at any point. In fact, Due to image rights restrictions, multiple line equations cannot be graphically displayed. = - ∞ and Due to image rights restrictions, multiple line equations cannot be graphically displayed. = + ∞ for every χ ∈ R. [ABSTRACT FROM AUTHOR]
The least squares derivative and the approximate derivative are both generalizations of the ordinary derivative. The existence of either of these generalized derivatives does not guarantee the existence of the other and it is even possible for both generalized derivatives to exist at a point but have different values. Several examples of such functions are presented in this paper. In addition, conditions for which the existence of the approximate derivative implies the existence (and equality) of the least squares derivative are stated and proved. These conditions involve the notion of Hölder continuity and a stronger version of approximate differentiability. [ABSTRACT FROM AUTHOR]