On Symmetric Riemann-Derivatives

The basic properties like monotoni city, Darboux property, mean value property of symmetric Riemann-derivatives of order n of a real valued function f at a point x of its domain (a closed interval) is studied. In some cases, function is considered to be continuous or semi-continuous.

On Symmetric Riemann-Derivatives

The basic properties like monotoni city, Darboux property, mean value property of symmetric Riemann-derivatives of order n of a real valued function f at a point x of its domain (a closed interval) is studied. In some cases, function is considered to be continuous or semi-continuous.

Weighted uniform density ideals

Weighted uniform densities are a generalization of the uniform density, which is also known as the Banach density. In this paper, we introduce the concept of weighted uniform density ideals and consider the topological complexity of these ideals as well as when they have certain analytical properties related to the ideal convergence of sequences and series. Furthermore, we prove some inequalities between different upper and lower weighted uniform densities and give the answer to the problem concerning the Darboux property of these densities.

Invariance of nonatomic measures on effect algebras

The present paper deals with invariance of nonatomic measures defined on effect algebras. Firstly, it is proved that if μ is a nonatomic and continuous probability measure defined on a σ-complete effect algebra L, then it satisfies para-Darboux property. Then, the invariance between continuous probability measures m and μ defined on a σ-complete effect algebra L is established when μ is nonatomic satisfying para-Darboux property on L.

Comparison of Some Families of Real Functions in Algebraic Terms

We compare families of functions related to the Darboux property (functions having the 𝒜-Darboux property) with family of strong Świątkowski functions using the notions of strong c-algebrability. We also compare families of functions associated with density topologies.

Properties of the division topology on the set of positive integers

In this paper, we examine properties of the division topology on the set of positive integers introduced by Rizza in 1993. The division topology on [Formula: see text] with the division order is an example of [Formula: see text]-Alexandroff topology. We mainly concentrate on closures of arithmetic progressions and connected and compact sets. Moreover, we show that in the division topology on [Formula: see text], the continuity is equivalent to the Darboux property.