Sequential definitions of continuity for real functions

摘要

A function f : R --> R is continuous at a point a if, given a sequence x = (x(n)), lim x = u implies that lim f (x) = f (x(n)). This definition can be modified by replacing lim with an arbitrary linear functional G. Generalizing several definitions that have appeared in the literature, we say that f : R --> R is G-continuous at a if G(x) = u implies that G(f (x)) = f (u). When G(x) = lim(n) n(-1) Sigma(k=1)(n) x(k), Buck showed that if a function f is G-continuous at a single point then f is linear, that is, f (u) = an + b for fixed a and b. Other authors have replaced convergence in arithmetic mean with A-summability, almost convergence and statistical convergence. The results in this paper include a sufficient condition for G-continuity to imply linearity and a necessary condition for continuous functions to be G-continuous, thereby generalizing several known results in the literature. It is also shown that, in many situations, the G-continuous functions must be either precisely the linear functions or precisely the continuous functions. However, examples are found where this dichotomy fails, which, in particular, leads to a counterexample to a conjecture of Spigel and Krupnik. [References: 31]