A Variation on Statistical Ward Continuity



A sequence \((\alpha _{k})\) of points in \(\mathbb {R}\), the set of real numbers, is called \(\rho \) -statistically convergent to an element \(\ell \) of \(\mathbb {R}\) if
$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{\rho _{n}}|\{k\le n: |\alpha _{k}-\ell |\ge {\varepsilon }\}|=0 \end{aligned}$$
for each \(\varepsilon >0\), where \(\varvec{\rho }=(\rho _{n})\) is a non-decreasing sequence of positive real numbers tending to \(\infty \) such that \(\limsup _{n} \frac{\rho _{n}}{n}<\infty \), \(\Delta \rho _{n}=O(1)\), and \(\Delta \alpha _{n} =\alpha _{n+1}-\alpha _{n}\) for each positive integer n. A real-valued function defined on a subset of \(\mathbb {R}\) is called \(\rho \)-statistically ward continuous if it preserves \(\rho \)-statistical quasi-Cauchy sequences where a sequence \((\alpha _{k})\) is defined to be \(\rho \)-statistically quasi-Cauchy if the sequence \((\Delta \alpha _{k})\) is \(\rho \)-statistically convergent to 0. We obtain results related to \(\rho \)-statistical ward continuity, \(\rho \)-statistical ward compactness, ward continuity, continuity, and uniform continuity. It turns out that the set of uniformly continuous functions coincides with the set of \(\rho \)-statistically ward continuous functions not only on a bounded subset of \(\mathbb {R}\), but also on an interval.


Summability Statistical convergent sequences Quasi-Cauchy sequences Boundedness Uniform continuity 

Mathematics Subject Classification

Primary: 40A05 Secondaries: 26A05 26A15 26A30 



The author would like to thank the referees for a careful reading and several constructive comments that have improved the presentation of the results.


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Copyright information

© Malaysian Mathematical Sciences Society and Universiti Sains Malaysia 2015

Authors and Affiliations

  1. 1.Faculty of Arts and SciencesMaltepe UniversityMaltepe, IstanbulTurkey

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