2026/02/15 by Açıkgöz. · 1 voice
Mathematics · #math.GN
We introduce statistically p-upward quasi-Cauchy sequences, defined by the condition limn→∞(1)/(n)|\k≤ n: xk - xk+p≥ε\|=0 for every ε>0, and develop the corresponding notions of compactness and continuity. We prove that a subset of ℝ is statistically p-upward compact if and only if it is bounded below, characterizing lower boundedness sequentially. Statistically p-upward continuity is shown to imply uniform continuity on below bounded sets. The function space SUCp(E) is a closed convex cone that fails to be a vector subspace -- distinguishing it from all previously studied sequential continuity spaces. We establish that every non-decreasing uniformly continuous function belongs to SUCp(E), use Weyl's equidistribution theorem to show sin x\notinSUCp(ℝ), prove a step-parameter hierarchy, and show that SUCp(E)∩ Cb(E) is nowhere dense in Cb(E). As an application, we develop a one-sided error control theory for function approximation, illustrated by Bernstein operators on a pharmacokinetic model. The inclusion relations among the continuity types studied and open problems are provided.