On the conditions for existence and continuity of fractional velocity

TL;DR

This paper explores the mathematical conditions under which fractional velocity, a generalization of the derivative based on fractional powers, exists and is continuous, linking it to H"older exponents and local function behavior.

Contribution

It establishes the existence and continuity conditions for fractional velocity, relates it to H"older exponents, and compares it with the Kolwankar-Gangal local fractional derivative.

Findings

Fractional velocity exists under specific H"older conditions.

Continuous fractional velocity of non-integer order vanishes.

Fractional velocity characterizes discontinuities in derivatives.

Abstract

H\"older 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 H\"older 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. Finally the equivalence with the Kolwankar-Gangal local fractional derivative is investigated.