A sequence (α k) of real numbers is called λ-statistically upward quasi-Cauchy if for every ε > 0 lim n→∞ 1 λn |{k ∈ I n : α k − α k+1 ≥ ε}| = 0, where (λ n) is a non-decreasing sequence of positive numbers tending to ∞ such that λ n+1 ≤ λ n + 1, λ 1 = 1, and I n = [n − λ 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 λ-statistically upward continuous if it preserves λ-statistical upward quasi-Cauchy sequences. λ-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 λ-statistical upward continuous on a λ-statistical upward compact subset of R.