2013/07/09 by Huseyin Cakalli, Cakalli, Huseyin
Mathematics · #26A15 #40A05 #40A30 #FOS: Mathematics #General Mathematics (math.GM) #math.GM #msc:26A15 #msc:40A05 #msc:40A30
paper · pdf · doi:10.48550/arxiv.1307.2418
25 pages. arXiv admin note: substantial text overlap with arXiv:1205.3674, arXiv:1103.1230, arXiv:1102.1531, arXiv:1305.0697
arxiv created 2013/07/09 · arxiv updated 2013/07/10
A real valued function f defined on a subset E of R, the set of real numbers, is statistically upward continuous if it preserves statistically upward half quasi-Cauchy sequences, is statistically downward continuous if it preserves statistically downward half quasi-Cauchy sequences; and a subset E of R, is statistically upward compact if any sequence of points in E has a statistically upward half quasi-Cauchy subsequence, is statistically downward compact if any sequence of points in E has a statistically downward half quasi-Cauchy subsequence where a sequence (xn) of points in R is called statistically upward half quasi-Cauchy if limn→∞(1)/(n)|\k≤ n: xk-xk+1≥ ε\|=0 is statistically downward half quasi-Cauchy if limn→∞(1)/(n)|\k≤ n: xk+1-xk≥ ε\|=0 for every ε>0. We investigate statistically upward continuity, statistically downward continuity, statistically upward half compactness, statistically downward half compactness and prove interesting theorems. It turns out that uniform limit of a sequence of statistically upward continuous functions is statistically upward continuous, and uniform limit of a sequence of statistically downward continuous functions is statistically downward continuous.