**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
- 54C30 Real-valued functions, See also {26-XX}
- 54C60 Set-valued maps, See also {26E25, 28B20, 47H04, 58C06}
- 54C65 Selections, See also {28B20}

**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*

**Upload:** 2000-07-07

**Update:** 2000-07-07

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