Conditions for continuity of fractional velocity and existence of fractional Taylor expansions

Abstract

Hölder functions represent mathematical models of nonlinear physical phenomena. This work investigates the general conditions of existence of fractional velocity as a localized generalization of ordinary derivative with regard to the exponent order. Fractional velocity is defined as the limit of the difference quotient of the function’s increment and the difference of its argument raised to a fractional power. A relationship to the point-wise Hölder exponent of a function, its point-wise oscillation and the existence of fractional velocity is established. It is demonstrated that wherever the fractional velocity of non-integral order is continuous then it vanishes. The work further demonstrates the use of fractional velocity as a tool for characterization of the discontinuity set of the derivatives of functions thus providing a natural characterization of strongly non-linear local behavior. A link to fractional Taylor expansions using Caputo derivatives is demonstrated.