SEQUENTIAL DEFINITIONS OF
CONTINUITY FOR REAL FUNCTIONS

JEFF CONNOR AND K.-G. GROSSE-ERDMANN

Abstract:

A function $f:{\bf R}\rightarrow{\bf R}$ is continuous at a point $u$ if, given a sequence ${\bf x}=( x_{n})$, $\lim{\bf x}=u$ implies that $\lim f({\bf x}) =f(u)$. 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 $f:{\bf R}\rightarrow{\bf R}$ is $G$-continuous at $u$ if $G( {\bf x}) =u$ implies that $G(f({\bf x}))=f(u)$. When $G({\bf x})=\lim_n n^{-1}\sum_{k=1}^{n}x_{k}$, Buck showed that if a function $f$ is $G$-continuous at a single point then $f$ is linear, that is, $f(u)=au+b$ 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.