论函数的可导与分数阶可导

从是否存在一点可导的相关函数和求导法则间相互关系的视角讨论函数的可导性问题，在分析一元分段函数在分界点处的导数问题的基础上，引进Riemann-Liouville分数阶导数定义和Caputo分数阶导数的定义，探讨分数阶导数与整数阶导数的相容性问题，研究分数阶可导问题。结果表明：仅在一点可导的函数及其他相关函数是存在的；导数的加法运算在四则运算中最为重要，复合函数的求导法在求导方法中最重要；Riemann-Liouville分数阶导数与经典整数阶导数具有相容性，Caputo分数阶导数与经典整数阶导数的相容性略差。

On the Differentiability and Fractional Order Differentiability of Functions

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 RiemannLiouville 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;RiemannLiouville fractional derivative is compatible with classical integer derivative,and Caputo fractional derivative is slightly incompatible with classical integer derivative.