On Differences of Semi-Continuous Functions

Abstract

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$_\alpha f$ 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 $\sum \phi_j$ of DSC functions on K converges to a DSC function provided $\sum{osc}_\alpha \phi_j$ converges locally uniformly for all countable ordinals $\alpha$.