试论积分第一中值定理

ON THE FIRST MEAN VALUE THEOREM OF INTEGRAL

以微积分基本定理为桥梁,利用实变函数论中的一些重要结果与函数逼近论中的Weierstrass第一定理及其Bernstein证明,在条件减弱的情形下,获得了比通常的积分第一中值定理更强的结论,且试图揭示积分第一中值定理与微分中值定理间深刻的联系.

With the fundamental theorem of calculus as a bridge, by using some important results from the function theory of real variables and Weierstrass＇s first theorem with Bernstein＇s proof in the approximation theory, some stronger results than the classical first mean value theorem of integral are obtained, which reveal the deep relation between the first mean value theorem of integral and the mean value theorems of differential.