Local and global properties of quasi-continuous and cliquish functions

Preprint series: 99-60, Reports on Analysis

Preprint series: , Reports on Analysis

MSC:
54C08 Weak and generalized continuity
41A30 Approximation by other special function classes

Abstract: We consider real-valued quasi-continuous and
cliquish functions $f$ on an arbitrary topological space
$X$. For characterizing their local properties we introduce
a multifunction on $X$, the open hull $HO_f(x)$, having
its values in the family of closed sets of real numbers.
The global claims about quasi-continuous and cliquish
functions concern approximation properties. They say that
every quasi-continuous function and, similarly, every
cliquish function can be represented as a uniform limit of
a sequence of step functions $\varphi_n$ defined on a chain
of so-called semi-open or almost semi-open partitions of
$X$, respectively.

Keywords: quasi-continuous function, neighborly function, semi-continuous function, robust function, cliquish function, semi-open set, robust set, full hull, open hull, admissible modification, semi-open partition, semi-open step function, almost semi-open partition, almost semi-open step function, uniform approximation, associated multifunction, lower (upper) semi-continuity, lower (upper) quasi-continuity