# A Variation on Statistical Ward Continuity

Article

## Abstract

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.

## Keywords

Summability Statistical convergent sequences Quasi-Cauchy sequences Boundedness Uniform continuity

## Mathematics Subject Classification

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

## Notes

### Acknowledgments

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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