数学分析理论中变上限积分问题的教学拓展研究

Teaching Expansion of the Problem of Variable Upper Bound Integral in the Theory of Mathematical Analysis

本文分别从f(x)的Riemann可积性和连续性出发,厘清了f(x)与其相对应变上限积分函数Φ(x)的连续性和可导性之间的局部内在联系,并首次给出了许多数学分析教材中常见的一个典型例题的新证明方法,最后讨论了上述理论的若干应用。

Starting from the Riemann integrability and continuity of f( x),the local internal relations between the integrability and the differentiability of its corresponding upper limit integral function Φ( x) and f( x) are clarified,and a new proof method of a typical example commonly used in many mathematical analysis textbooks is given for the first time. Some applications of the above theory are discussed.