On an Example of a Function with a Derivative which does not have a Third Order Symmetric Riemann Derivative Anywhere

Abstract

In this paper we construct a differentiable function \( F : \mathbb{R} \to \mathbb{R} \) that does not have a third order symmetric Riemann derivative at any point. In fact, \[ \underline{SRD}^3F(x) = \liminf_{h \to 0} \tfrac{F(x + 3h) - 3F(x+h) + 3F(x - h) - F(x - 3h)}{(2h)^3} = - \infty \] and \[ \overline{SRD}^3F(x) = \limsup_{h \to 0 }\tfrac{F(x + 3h) - 3F(x+h) + 3F(x - h) - F(x - 3h)}{(2h)^3} = + \infty \] for every \( x \in \mathbb{R}. \)