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.

 

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How to cite this article:

Zeinab Bandpey and Bhamini M. P. Nayar, A study of generalized continuous functions, International Journal of Advances in Mathematics, Volume 2018, Number 6, Pages 16-26, 2018.