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
(xnβ) of points in R is called statistically upward half
quasi-Cauchy if nββlimβn1ββ£{kβ€n:xkββxk+1ββ₯Ξ΅}β£=0 is statistically downward half
quasi-Cauchy if nββlimβn1ββ£{kβ€n:xk+1ββxkββ₯Ξ΅}β£=0 for every Ξ΅>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.Comment: 25 pages. arXiv admin note: substantial text overlap with
arXiv:1205.3674, arXiv:1103.1230, arXiv:1102.1531, arXiv:1305.069