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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 $0\leqx<1$ and \[\lim_{x \to 1^{-}}(1-x) \sum_{k=0}^{\infty}p_{k}x^{k}=\ell.\] Weintroduce a concept of Abel continuity in the sense that a function $f$ definedon a subset of $\Re$, the set of real numbers, is Abel continuous if itpreserves Abel convergent sequences, i.e. $(f(p_{n}))$ is an Abel convergentsequence whenever $(p_{n})$ is. A new type of compactness, namely Abelsequential compactness is also introduced, and interesting theorems related tothis kind of compactnes, Abel continuity, statistical continuity, lacunarystatistical continuity, ordinary continuity, and uniform continuity areobtained.

Abel Continuity