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\author
{Huseyin Cakalli}
\address{Maltepe University, Department of Mathematics, Marmara Egitim Koyu, TR34857, Maltepe,�stanbul, Turkey}
\email{hcakalli@maltepe.edu.tr}

\begin{document}
\title
{FORWARD COMPACTNESS}


\thanks{}
\subjclass[2000]{Primary: 26A15; Secondary: 40A05 } \keywords{Real functions, sequences, compactness}%up to five keywords




\begin{abstract}


A real function $f$ is continuous if and only if $(f(x_{n}))$ is a convergent sequence whenever $(x_{n})$ is convergent and a subset $E$ of $\textbf{R}$ is compact if any sequence $\textbf{x}=(x_{n})$ of points in $E$ has a convergent subsequence whose limit is in $E$ where $\textbf{R}$ is the set of real numbers. These well known results suggest us to introduce a concept of forward continuity in the sense that a function $f$ is forward continuous if  $\lim_{n\rightarrow\infty} \Delta f(x_{n})=0$ whenever $\lim_{n\rightarrow\infty} \Delta x_{n}=0$ and a concept of forward compactness in the sense that a subset $E$ of $\textbf{R}$ is forward compact if any sequence $\textbf{x}=(x_{n})$ of points in $E$ has a subsequence $\textbf{z}=(z_{k})=(x_{n_{k}})$ of the sequence $\textbf{x}$ such that $\lim_{k\rightarrow \infty} \Delta z_{k}=0$ where $\Delta y_{k}=y_{k+1}-y_{k}$.
We prove that any forward continuous function defined on a forward compact subset of $\textbf{R}$ is uniformly continuous, uniform limit of forward continuous functions is forward continuous, any forward continuous function is continuous. Some other results related to forward continuity and continuity are also obtained and some open problems are discussed.
\end{abstract}

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\begin{thebibliography}{2}
\bibitem{Cakalli1} Cakalli, H., Sequential definitions of compactness, Applied Mathematics Letters, 21 , No:6,
594-598, (2008)
\bibitem{Cakalli2} Cakalli, H., Slowly oscillating continuity, Abstract and Applied Analysis, Volume 2008,
Article ID 485706


\end{thebibliography}
\end{document}

