On λ Statistical Upward Compactness and Continuity

A sequence (alpha(k)) of real numbers is called lambda-statistically upward quasi-Cauchy if for every epsilon > 0 lim(n ->infinity)1/lambda(n)vertical bar{K is an element of I-n : alpha(k) - alpha(k+1) >= epsilon}vertical bar = 0, where (lambda(n)) is a non decreasing sequence of positive numbers tending to infinity such that lambda(n+1) <= lambda(n) + 1, lambda(1) = 1, and I-n = [n - lambda(n) + 1,n] for any positive integer n. A real valued function f defined on a subset of R, the set of real numbers is lambda-statistically upward continuous if it preserves lambda-statistical upward quasi-Cauchy sequences. lambda-statistically upward compactness of a subset in real numbers is also introduced and some properties of functions preserving such quasi Cauchy sequences are investigated. It turns out that a function is uniformly continuous if it is A-statistical upward continuous on a lambda-statistical upward compact subset of R.