闭区间上连续函数有界性的教学研究与应用

Research on teaching of the boundedness of continuous function defined in close interval and its application

大学数学专业课程《数学分析》是一门非常重要的专业基础课和入门课程,闭区间上连续函数的性质是该课程的重要教学内容.关于闭区间上连续函数的有界性定理,该文给出一个新的完全不同的证明思路.从局部出发渐变到整体,将局部性质推演为整体性质,是新证明的出发点和入手点.该证明思路的核心是确界原理的应用,并将此新的证明思路应用于研究连续函数的其他性质,如连续函数的相邻的两个最值点区间的确定、连续函数的介值定理等.

In this paper,we study the properties of continuous function defined in closed interval which is one of most important knowledge points.The course Mathematical Analysis is one of most important one in Department of Mathematics.We will give a new proof of boundedness of continuous function by using principle of supremum and infimum.Also,this new method of proof will be applied to prove other important properties of continuous function defined in closed interval.