# Local Fractional Derivatives of Differentiable Functions are Integer-order Derivatives or Zero

## TL;DR

This paper proves that local fractional derivatives of differentiable functions are either integer-order derivatives or zero, indicating they are not suitable for describing fractal or nowhere differentiable functions.

## Contribution

It establishes that local fractional derivatives are limited to integer orders or zero, contrasting with fractional derivatives like Caputo's that can handle non-integer orders.

## Key findings

- Local fractional derivatives of differentiable functions are either integer-order derivatives or zero.
- Local fractional derivatives cannot describe nowhere differentiable functions or fractals.
- Leibniz rule does not hold for derivatives of non-integer order.

## Abstract

In this paper we prove that local fractional derivatives of differentiable functions are integer-order derivative or zero operator. We demonstrate that the local fractional derivatives are limits of the left-sided Caputo fractional derivatives. The Caputo derivative of fractional order alpha of function f(x) is defined as a fractional integration of order (n-alpha) of the derivative f^(n)(x) of integer order n. The requirement of the existence of integer-order derivatives allows us to conclude that the local fractional derivative cannot be considered as the best method to describe nowhere differentiable functions and fractal objects. We also prove that unviolated Leibniz rule cannot hold for derivatives of orders alpha, which are not equal to one.