再议函数介值性

Reconsider the Intermediate Value Property of Functions

本文首先将高中数学教材中关于区间的九个规定抽象为精准的数学定义,指出函数介值性的本质特征就是函数的值域集合为区间,并给出了函数介值性两个定义的等价性证明.同时讨论了函数具有介值性的充分条件及具有介值性的函数与连续函数的关系,并给出了函数介值性在解决微积分问题中的若干应用.本文的结论厘清了区间、函数介值性与函数连续性定义之间的相互关系,为做好中学数学与大学数学的衔接提供了有益的参考和支撑.

This paper introduces the nine provisions on intervals in senior middle school math textbook,abstract them into precise mathematical definitions,and point out that the essential characteristic of the intermediate value of function is that the range set of a function is an interval.The equivalence of two definitions about intermediate value is proved.The sufficient conditions for functions with intermediate value and the relationship between functions with intermediate value and continuous functions are discussed,and some applications in solving Calculus problems are given.The relationship among interval,intermediate value and continuity seems cleared,which provides a useful reference and support for the connection between middle school and college mathematics.