C{k,1} Functions and Riemann Derivatives

Abstract

In this work we provide a characterization of $C^{k,1}$ functions of one real variable (that is, $k$ times differentiable with locally Lipschitz $k$-th derivative) by means of $(k+1)$-th divided differences and Riemann derivatives. In particular we prove that the class of $C^{k,1}$ functions is equivalent to the class of functions with bounded $(k+1)$-th divided difference. From this result we deduce a Taylor's formula for this class of functions and a characterization through Riemann derivatives.