关于Riemann可积函数的本性

On Essential Feature of Kiemann-Integrable Functions

本文引进了函数在一点的本性振幅的概念，在Ｒｉｅｍａｎｎ积分意义下，证明了定理：设有界函数ｆ定义于闭矩形Ｉ，在Ｉ上Ｒｉｅｍａｎｎ可积的充要条件是对任意η大于零，Ｅ＿η是一个零面积集。

In this paper,we introduce the concept for essential oscillation of a functions at a point.The following theorem is proved in the manner for Riemann integral.Theorem. Assume a function f is defined and bounded on a closed rectangle I=[a,b]×[c,d].The necessary and sufficient condition for f to be Riemann-integrable on I is that area of Eη={M|M∈I and ω[f,M]≥η} is zero for any η>0.