关于Riemann积分的若干注记

Some Notes on Riemannian Integral

本文指出了Riemann积分定义中分法与选点的二重任意性可改为一重任意,着重论述了Riemann可积与有原函数没有必然的蕴涵关系,建立了导函数可积的充要条件,举例说明了两个可积函数的复合函数不一定可积.

This paper points out that the double arbitrariness of the dividing method and point selection in the definition of Riemannian integral can be changed into single arbitrariness, discusses the implication relation that there is no certainty between Riemannian integrability and original function, establishes the necessary and sufficient condition of derived function integrability, and states with examples that the compound function of two integrable functions is uncertainly integrable.