连续函数介值定理及其推论的逆命题

THE CONVERSE INTERMEDIATE VALUE THEOREM OF CONTINUOUS FUNCTIONS AND ITS COROLLARY

本文利用介值定理的结论引进了一个函数是区间保持的概念,并给出了关于区间保持的函数是连续的一个充要条件。我们证明了对于单调函数,区间保持的性质与连续性等价。最后还证明了对于实数的子集A,如果定义在A上取值于A内的任一连续函数都有不动点,则A为实数的有限闭区间。

In this paper we introduce the concept that a function is intervalpreserving by the assertion of intermediate value theorem and give a necessary and sufficient condition for the continuity of an interval-preserving func tion. We prove that interval-preserving property is equivalent to continuity for a monotone function. And finally we prove that let A be a subset of real numbers,if every continuous function which is defined on A and taken value in A has fixed point, then A is a finite closed interval.