  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.
