迭代修复含有限个同类型间断函数的连续性

Continuity of Functions with Finitely Many Discontinuities of the Same Type Repaired by Iteration

我们已证明具有一个间断点的函数有连续的二次迭代.它实际上表明在迭代之下它的间断点能被自己函数对修复为连续点.如果一个函数含至少两个间断点,那么,在迭代之下,它的间断点或者被它自己函数对修复为连续点或者被其它间断点的函数对修复为连续点.本文研究具有多于一个但是只含有限个同类型间断点的不连续函数,给出了这些函数二次迭代连续的充分必要条件.

It was proved that a function with exact one discontinuity may have a continuous iterate of second order.It actually shows that its discontinuity may be repaired to be a continuous one by its own pair of functions under iteration.If a function has at least two discontinuities,then each of its discontinuities may be repaired to be a continuous one by either its own pair of functions or the other’s pair of functions under iteration.In this paper we investigate those functions having more than one but finitely many discontinuities of the same type and give necessary and sufficient conditions for those functions whose second order iterates are continuous.