On continuity properties of the improved conformable fractional derivatives

Abstract

The conformable fractional derivative has been introduced to extend the familiar limit definition of the classical derivative. Despite having many advantages compared to other fractional derivatives such as satisfying nice properties as classical derivative and easy to solve numerically, it also has disadvantages as it gives large error compared to Riemann-Liouville and Caputo fractional derivatives. Modified types of conformable derivatives have been proposed to overcome the shortcoming. The improved conformal fractional derivatives are declared to be better approximations of Riemann-Liouville and Caputo derivatives in terms of physical behavior. In this paper, properties concerning continuity of the improved conformable fractional derivative are investigated. We prove the relation between -differentiable and continuity of a function and corresponding interior extremum theorem. We also prove the properties close to Rolle’s Theorem and Mean Value Theorem for the improved conformable fractional derivatives.