2012/05/14 by Hüseyi̇n Çakallı, Huseyin Cakalli, Cakalli, Huseyin · 2 citations
Mathematics · #40A30 #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #General Mathematics (math.GM) #Primary: 40A05 #Rings, Modules, and Algebras #Secondaries: 26A15 #math.GM #msc:26A15 #msc:40A05 #msc:40A30
paper · pdf · doi:10.48550/arxiv.1205.3674
24 pages. arXiv admin note: substantial text overlap with arXiv:1103.1230, arXiv:1102.1531, arXiv:1204.2445
arxiv created 2012/05/14 · openalex publication_date 2012/05/14 · arxiv updated 2012/05/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
A real function f is ward continuous if f preserves quasi-Cauchyness, i.e. (f(xn)) is a quasi-Cauchy sequence whenever (xn) is quasi-Cauchy; and a subset E of R is quasi-Cauchy compact if any sequence x=(xn) of points in E has a quasi-Cauchy subsequence where R is the set of real numbers. These known results suggest to us introducing a concept of upward (respectively, downward) half quasi-Cauchy continuity in the sense that a function f is upward (respectively, downward) half quasi-Cauchy continuous if it preserves upward (respectively, downward) half quasi-Cauchy sequences, and a concept of upward (respectively, downward) half quasi-Cauchy compactness in the sense that a subset E of R is upward (respectively, downward) half quasi-Cauchy compact if any sequence of points in E has an upward (respectively, downward) half quasi-Cauchy subsequence. We investigate upward(respectively, downward) half quasi-Cauchy continuity and upward (respectively, downward) half quasi-Cauchy compactness, and prove related theorems.