分段点处高阶导数存在性的直接判别法

Criteria for the Existence of Higher Derivatives of Piecewise Functions

利用泰勒展开式,把分段函数在以x0为分段点的相邻区间上展开为(x-x0)的泰勒级数,然后通过观察和比较相邻分段区间上泰勒展开式中的k阶系数是否对应相等,来判断分段函数在x=x0点是否存在k阶导数.

Using Taylor expansion, a piecewise-defined function can be expanded as power series of （x-xo） in a neighborhood of x0, one for x〉x0 and the other for x〈x0. By comparing the corresponding Taylor coefficients we can establish criteria for the existence of higher derivatives of the function at x = x0.