Cliquish Functions Research Articles

Point-set games and functions with the hereditary small oscillation property

Given a metric space X, we consider certain families of functions f:X→R having the hereditary oscillation property HSOP and the hereditary continuous restriction property HCRP on large sets. When X is Polish, among them there are families of Baire measurable functions, μ‾-measurable functions (for a finite nonatomic Borel measure μ on X) and Marczewski measurable functions. We obtain their characterizations using a class of equivalent point-set games. In similar aspects, we study cliquish functions, SZ-functions and countably continuous functions.

On Semi-Continuous and Clisquish Functions in Generalized Topological Spaces

In this paper, we will focus on three types of functions in a generalized topological space, namely; lower and upper semi-continuous functions, and cliquish functions. We give some results for nowhere dense sets and for second category sets. Further, we discuss the nature of cliquish functions in generalized metric spaces and provide the characterization theorem for cliquish functions in terms of nowhere dense sets.

On Density and Sigma-Porosity in Some Families of Darboux Functions

A cliquish function $f: \mathbb R\to\mathbb R$ is internally cliquish if its set of discontinuity points is nowhere dense. A Baire $1$ function $f: \mathbb R\to\mathbb R$ is internally Baire $1$ if its set of discontinuity points is nowhere dense. In this paper we proved that the set of Darboux internally cliquish (Baire $1$) functions is dense and $\sigma$-strongly porous in the family of Darboux cliquish (Baire $1$) functions.

On one-sidedly graph cliquish functions

On one-sidedly graph cliquish functions

A note on graph quasi-continuous and graph cliquish functions

In this paper we study under which conditions there is exactly one quasi-continuous function whose graph is contained in the closure of the graph of a graph quasi-continuous function. Also, we study the relation between a graph cliquish function and a cliquish function whose graph is contained in the closure of the graph of the graph cliquish function.

On graph cliquish functions

In the present paper we introduce a new notion of graph cliquish functions from a topological space to a metric space and study its relation with other types of generalized continuity. We also give a characterization of that new notion of generalized continuity on a dense set of points.

On lower and upper semi-continuous functions

We give various relations between lower (upper) semi-continuous functions and cliquish functions in a generalized topological space. Also, we answer the open problems raised by Korczak-Kubiak et al. in [5].

Dense and σ-porous subsets in some families of cliquish functions

A function f: ℝ → ℝ is internally cliquish (internally quasi-continuous) if it is cliquish (quasi-continuous) and the set of discontinuity points of f is nowhere dense. We prove that the set of internally cliquish (internally quasi-continuous) functions is dense and σ-strongly porous set in the family of cliquish (quasi-continuous) functions. The analogous result is obtained also for the family of functions having the Świątkowski property.

Porous subsets in the space of functions having the Baire property

Abstract The comparison of some subfamilies of the family of functions on the real line having the Baire property in porosity terms is given. We prove that the family of all quasi-continuous functions is strongly porous set in the family of all cliquish functions and that the family of all cliquish functions is strongly porous set in the family of all functions having the Baire property. We prove also that the family of all Świątkowski functions is lower 2/3-porous set in the family of cliquish functions and the family of functions having the internally Świątkowski property is lower 2/3-porous set in the family of cliquish functions.

Separating sets by cliquish functions

In this paper we characterize the pairs 〈A0,A1〉 of disjoint subsets of topological space X which can be separated by a cliquish function.

Quotients of cliquish functions on some interval or on no interval

We study cardinal invariants connected with quotients in the case of functions that are constant, continuous, quasi-continuous, and cliquish on some interval or on no interval.

Some continuous operations on pairs of cliquish functions

Abstract The algebraic or lattice operations in the classes of cliquish or quasicontinuous functions are well known [Z. Grande: On the maximal multiplicativefamily for the class of quasicontinuous functions, Real Anal. Exchange 15 (1989-1990), 437-441, Z. Grande, L. Soltysik: Some remarks on quasicontinuousreal functions, Problemy Mat. 10 (1990), 79-86]. This also pertains to the symmetrical quasicontinuity or symmetrical cliquishness [Z. Grande: On the maximaladditive and multiplicative families for the quasicontinuities of Piotrowskiand Vallin, Real Anal. Exchange 32 (2007), 511-518]. In this article, we examine the superpositions F(f, g), where F is a continuous operation and f, g are cliquish (symmetrically cliquish) or f is continuous (f is symmetrically quasicontinuous with continuous sections) and g is quasicontinuous (symmetrically quasicontinuous).

Some Observations on the Symmetrical Quasicontinuity of Piotrowski and Vallin

In this article I investigate the measurability of symmetrically quasicontinuous (in the sense of Piotrowski and Vallin) functions and symmetrically cliquish functions $f : R^2\to R$ with respect to the Euclidean topology and with respect to the density topology $T_d\times T_d$.

Cluster Sets and Approximation Properties of Quasi-Continuous and Cliquish Functions

Cluster Sets and Approximation Properties of Quasi-Continuous and Cliquish Functions

Characterization of Cliquish Functions

For a real function f defined on a perfect Baire space X the following are equivalent: f is cliquish, and f is a pointwise limit of two monotone sequences of upper and lower quasi-continuous functions, respectively.

On theorems of Pu & Pu and Grande

Given a finite family of cliquish functions, $\A$, we can find a Lebesgue function $\alpha$ such that $f+ \alpha$ is Darboux and quasi-continuous for every $f \in\A$. This theorem is a generalization both of the theorem by H. W. Pu & H. H. Pu and of the theorem by Z. Grande.

On the sums and the products of Darboux cliquish functions

On the sums and the products of Darboux cliquish functions

Note on limits of simply continuous and cliquish functions

The main result of this paper is that any function f defined on a perfect Baire space (X, T) with values in a separable metric space Y is cliquish (has the Baire property) iff it is a uniform (pointwise) limit of sequence {fn : n ≥ 1} of simply continuous functions. This result is obtained by a change of a topology on X and showing that a function f : (X, T) → Y is cliquish (has the Baire property) iff it is of the Baire class 1 (class 2) with respect to the new topology.

On Limits of Quasicontinuous, Simply Continuous and Cliquish Functions

On Limits of Quasicontinuous, Simply Continuous and Cliquish Functions

On quasicontinuous and cliquish functions

On quasicontinuous and cliquish functions