A study of generalized continuous functions -by Zeinab Bandpey and Bhamini M. P. Nayar

# A study of generalized continuous functions-by Zeinab Bandpey and Bhamini M. P. Nayar

Abstract: In the paper, Weak Continuity Forms, Graph Conditions and Applications, the concept of $u$-continuous functions are introduced and presented several applications of such functions. In the present study, by generalizing the concept of $u$-continuity using the notion of an $\alpha$-set, introduced by O. Njastad , three classes of functions are introduced and studied.The concepts introduced here are strongly $u$-continuous functions, $\alpha u$-continuous functions and semi-$\alpha u$- continuous functions. A function $g: X \rightarrow Y$ is $\alpha u$-continuous (strongly $u$-continuous, semi-$\alpha u$-continuous) at $x \in X$, if for each $\alpha$-set ($\alpha$-set, open set) $W$ which contains a closed neighborhood of $g(x)$, there exists an $\alpha$-set (open set, $\alpha$-set ) $V$which contains a closed neighborhood of $x$ and satisfies condition $g (clV)\subseteq clW$. If $g$ is $\alpha u$-continuous (strongly $u$-continuous, semi-$\alpha u$-continuous) at each $x\in X$, we say $g: X \rightarrow Y$ is $\alpha u$-continuous (strongly u-continuous, semi-$\alpha u$-continuous) on $X.$

Keywords: $\alpha$-set, $u$-continuity, $\alpha$-closure.