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On the Differentiability and Fractional Order Differentiability of Functions

LIU Shuxia;ZHAN Huashui;School of Mathematics and Statistics, Xiamen University of Technology;  
This paper discusses the differentiability of function from the perspective of whether there is a differentiable correlation function and the relationship between derivation rules.Based on the analysis of the derivative of univariate piecewise function at the boundary point, the definitions of Riemann-Liouville fractional derivative and Caputo fractional derivative are introduced to discuss the compatibility between fractional derivative and integer derivative, and the fractional derivative problem is studied.The results show that functions and other related functions that can be derived only at one point exist; the addition operation of derivative is the most important in the four operations, and the derivation method of composite function is the most important in the derivation method; Riemann-Liouville fractional derivative is compatible with classical integer derivative, and Caputo fractional derivative is slightly incompatible with classical integer derivative.
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