分段函数分段点可导性的一个定理及应用

A Theorem of Derivable Piecewise Function Separation and Its Application

给出分段函数分段点导数存在的一个充要条件:函数在该点连续,导函数在该点左、右极限存在且相等。并由此得到在分段点导数不存在的一个充分条件以及三种特例分段函数分段点导数存在的充分条件。举例说明该定理的应用,并指出利用该定理求分段函数分段点导数时的几点注意:函数在该点连续是可导的必要条件,导函数在该点左、右极限存在且相等是充分条件。

In this article, a sufficient and necessary condition under which the derivative of piecewise function separation exists is given： the function at this point is continuous, and the derivative exists at the left and right limits and is equal. As a result, a sufficient condition of non - existence of piecewise point and the one of ex- istence of piecewise derivative of three special cases piecewise function are obtained here. And the application of this theorem is illustrated through examples. Meanwhile, it＇s pointed out that attention should be paid to the solution to the separation derivative of piecewise function with this theorem.