反函数连续性和可微性的若干注记

Some Remarks on the Continuity and Differentiability of Inverse Function

在不要求函数在区间连续的假设下,研究了其反函数存在的条件及其在一点的连续和可微的条件,给出了反函数在一点连续的本质刻画.主要结论是原函数在某点连续不是其反函数在相应点连续的必要条件,而是函数将区间映射为区间,最后用例子说明结论的直观性.

In this paper, we study the conditions for the existence, the continuity and the differentiability of inverse functions without the assumption of interval continuity, and characterize the point continuity of an inverse function. Our main result is that the continuity of a function at a point is not necessary to the continuity of its inverse at the corresponding point, but the interval-into-interval mapping of a function.