广义一致收敛与亚一致收敛

Generalized Uniform Convergence and Hypouniform Convergence

本文讨论了连续函数列{f_2(x)}的极限函数f(x)连续的条件。采用了先把{f_2(x)}为正则收敛的条件减弱为弱正则收敛,或减弱为一致收敛,再减弱为广义一致收敛,最后成为一个定理:在[a,b]上的连续函数列{f_n(x)}的极限函数f(x)连续的充要条件是{f_n(x)}在[a,b]上是亚一致收敛的。

Let {fn(x)} be a sequence of continuous functions and f(x) be its limit. The condition forf(x) to be continuous is studied. First, the regular convergence restriction of {fa(x)} is relaxed to the requirement of weak regular convergence or uniform convergence of the sequence. Second, the condition is further weakened to generalized uniform convergence. Finally, a theorem is established : Suppose that {fa(x )} is a sequence of continuous functions on [a ,b] and / (z ) is its limit function. Then the necessary and sufficient condition for f(x) to be continuous is that {fa(x)} is hypouniformly convergent on [a,b].