关于半连续函数的差分
On Differences of Semi-Continuous Functions
摘要：
给出了Polish空间K上半连续函数的差类DSC（K）的外、内刻画，并利用限制于模糊集的性质给出了DSC（K）和K上点态稳定函数类PS（K）的分解刻画.主要的，外在的特征给出的行为限制在任何闭子集K的第二类的一些子集。利用第二作者定义的函数$f$的超限振荡osc $_αf$，引入了强连续点的概念.主要的内在表征产生以下DSC类似的Baire的表征第一Baire类函数：一个函数属于DSC（K）当且仅当其限制到任何封闭的非空集L有一个强连续点。作为推论，给出了K上DSC函数的局部一致收敛级数$\sum ϕ_j$收敛于一个DSC函数，只要$\sum{osc}_αϕ_j$对所有可数序数$α$局部一致收敛.
英文摘要：
Extrinsic and intrinsic characterizations are given for the class DSC(K) of differences of semi-continuous functions on a Polish space K, and also decomposition characterizations of DSC(K) and the class PS(K) of pointwise stabilizing functions on K are obtained in terms of behavior restricted to ambiguous sets. The main, extrinsic characterization is given in terms of behavior restricted to some subsets of second category in any closed subset of K. The concept of a strong continuity point is introduced, using the transfinite oscillations osc$_αf$ of a function $f$ previously defined by the second named author. The main intrinsic characterization yields the following DSC analogue of Baire's characterization of first Baire class functions: a function belongs to DSC(K) iff its restriction to any closed non-empty set L has a strong continuity point. The characterizations yield as a corollary that a locally uniformly converging series $\sum ϕ_j$ of DSC functions on K converges to a DSC function provided $\sum{osc}_αϕ_j$ converges locally uniformly for all countable ordinals $α$.
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