  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 $\alpha$.

Functional Analysis