关于和函数连续性教学的进一步思考

级数是产生新函数的重要方法,是研究函数的重要工具,是分析学的重要组成部分.随着级数理论的完善与发展,人们逐渐发现,函数项级数和函数的连续性这一分析性质非常重要而且应用十分广泛.一致收敛正是为了深入研究和函数的分析性质而引入的,然而在教学中我们发现,一致收敛性是很苛刻的,它只是保证和函数拥有良好分析性质的充分条件,但不是必要条件.事实上,保证和函数拥有连续性质的条件还可以适当减弱,本文正是从这一点出发,探索出了保证函数项级数的和函数连续性的弱化条件.

A Further Consideration on the Continuity of Sum Function

It is known that series is an important part in analytics, and is also a vital method to gen-erate new functions and a significant tool to study functions. With the development and improvement of series theory, people gradually find that the continuity of sum function of series of functions is very impor-tant and useful. The introduction of uniform convergence is to further research analysis nature of sum function. However, the uniform convergence is very demanding, which is the sufficient condition to ensure good analysis nature of sum function, but not the necessary one. Thus the condition guaranteeing the sum function having good analytic property can be weakened appropriately in order to explore the weakening condition of continuity of sum function of series of functions. From this point, firstly, we give several con-vergence concepts weaker than uniform convergence. Secondly, we give four continuity theorems of sum function of series of functions. Moreover, there are corresponding examples for these theorems. Finally, we acquire the weakening conditions.