Sequential Definitions of Continuity for Real Functions

Abstract

A function f: R → R is continuous at a point u if, given a sequence x = (x n ), lim x = u implies that lim f(x) = f(u). 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 u if G(x) = u implies that G(f(x)) = f(u). When G(x) = lim n n -1 Σ n k=1 x k , Buck showed that if a function f is G-continuous at a single point then f is linear, that is, f(u) = au+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.