LACUNARY STATISTICALLY UPWARD AND DOWNWARD HALF QUASI-CAUCHY SEQUENCES
DOI:
https://doi.org/10.5281/85fvwv28Keywords:
Summability, lacunary statistical convergence, boundedness and continuityAbstract
Abstract. A real valued function defined on a subset E of R, the set of real numbers, is lacunary
statistically upward continuous if it preserves lacunary statistically upward half quasi-Cauchy
sequences where a sequence (xk) of points in R is called lacunary statistically upward half quasiCauchy if
limr→∞
1
hr
|{k ∈ Ir : xk − xk+1 ≥ ε}| = 0
for every ε > 0; and (xk) is called lacunary statistically downward half quasi-Cauchy if
limr→∞
1
hr
|{k ∈ Ir : xk+1 − xk ≥ ε}| = 0
for every ε > 0, where θ = (kr) is an increasing sequence of non-negative integers such that k0 = 1
and hr : kr − kr−1 → ∞. We investigate lacunary statistically upward continuity and lacunary
statistically downward continuity and prove some interesting theorems. It turns out that not
only a lacunary statistically upward continuous function on a below bounded subset, but also
a lacunary statistically downward continuous function on an above bounded subset is uniformly
continuous.
References
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