中文摘要：

为研究函数在不可微处的局部行为,各种局部分数阶微分定义被提出,α-微分是其中重要的一种。本文研究了α-微分的一些性质,证明了利用α-微分研究函数局部行为的合理性和α-微分的几何意义的合理性。当f（x）连续α-可微时（0〈α〈1）,对于求解f（x）=0,作者提出了局部分数阶牛顿法且当f（α）（x）满足指数为α（12〈α〈1）的H？lder条件时,该算法是局部 Q-超线性收敛的。

英文摘要：

To study the behavior of a function at non-differentiable point,various definitions of local fractional derivative were proposed,α-derivative is a very important one.In this paper,some properties ofα-derivative are presented,the rationality of the method that study local behavior of a function by means ofα-derivative is demonstrated,and the rationality of the geometric meaning ofα-derivative is verified.It is of great importance to solve f（x）= 0 when f（x）is nowhere differentiable.For continu-ouslyα-differentiable function （0〈α〈1）,local fractional Newton method is proposed.The algorithm is locally Q-superlinearly convergent if f（α）（x）satisfies H？lder condition with exponentαwhere 1/2 〈α〈 1.