Abel Continuity

Abstract

A sequence $\textbf{p}=(p_{n})$ of real numbers is called Abel convergent to $\ell$ if the series $\Sigma_{k=0}^{\infty}p_{k}x^{k}$ is convergent for $0\leq x<1$ and \[\lim_{x \to 1^{-}}(1-x) \sum_{k=0}^{\infty}p_{k}x^{k}=\ell.\] We introduce a concept of Abel continuity in the sense that a function $f$ defined on a subset of $\Re$, the set of real numbers, is Abel continuous if it preserves Abel convergent sequences, i.e. $(f(p_{n}))$ is an Abel convergent sequence whenever $(p_{n})$ is. A new type of compactness, namely Abel sequential compactness is also introduced, and interesting theorems related to this kind of compactnes, Abel continuity, statistical continuity, lacunary statistical continuity, ordinary continuity, and uniform continuity are obtained.