关于定积分定义及可积条件的讨论

教材在定积分的定义中。通常特别强调分法和选取的两个任意性。利用此定义讨论问题比较繁琐．因此在此定义基础上做出进一步讨论．即将定义中分法的任意性用特殊分割来代替．这样不仅给解决问题带来了方便．而且可以帮助大家更深刻地理解和掌握定积分定义的实质。同时还给出了一个可积的充分必要条件．即关于函数收敛的Cauchy收敛准则．但应用此定理证明关于函数的可积性问题相当麻烦．由此我们对该定理做出改进．得出一个简单易行的判定定理。

The Definition of Definite Integral and Discussion of the Condition of an Integrable Function

The textbooks stress the arbitrary choice between the partition T and in the definition of definite integral,but it seems cumbersome to discuss issues with this definition .Therefore, this paper replace the partition T with a special partition in the definition on the base of further discussions.This method is not only convinent to settle issues in practice,but also helpful for the reader to gain a deep understanding and master the essence of the definite integral.Meanwhile,this paper gives an integrable necessary conditions--the Cauchy criterion for convergence.At last,this paper makes a simple theorem with improvements because the fact that the application of the definition to prove the integrable issues of a function appears cumbersome.