1999/01/28 by Fouad Chaatit, Chaatit, Fouad, Haskell P. Rosenthal +1
Mathematics · #04A15 #46B03 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical and Theoretical Analysis #Primary 26A21 #Secondary 03E15 #math.FA #msc:03E15 #msc:04A15 #msc:26A21 #msc:46B03
paper · pdf · doi:10.48550/arxiv.math/9901134
20 pages, AMSTeX
openalex publication_date 1999/01/28 · arxiv created 1999/02/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 oscαf 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 ∑ ϕj of DSC functions on K converges to a DSC function provided ∑oscαϕj converges locally uniformly for all countable ordinals α.