Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 Articles in Press 2026 06 06 Generalization of continuity 107064 10.48308/cgasa.2026.243504.1604 EN Omid Molaei Faculty of Mathematics and Computer Science, Hakim Sabzevari University, Sabzevar, Iran Ali Akbar Estaji Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. Mehdi Vatandoost Faculty of Mathematics and Computer Science, Hakim Sabzevari University, Sabzevar, Iran Journal Article 2026 02 09 This paper aims to redefine the concept of ``continuity of a function at a point'' from a set-theoretic perspective, providing a sufficiently flexible definition that encompasses the various forms of continuity found in the mathematical literature. Let $\mathscr{A}$ and $\mathscr{B}$ denote families of subsets of non-empty sets $X$ and $Y$, respectively. We define an $\mathscr{A}$-$ \mathscr{B}$-continuous map and examine some algebraic properties of structures related to the set $C_{(\mathscr{A}, \mathscr{B})}(X, Y)$, which consists of all $\mathscr{A}$-$ \mathscr{B}$-continuous maps from $X$ to $Y$. Additionally, we show that $C_{_{(\mathscr{A}, \mathcal{O}Y)}}(X, Y)\cong C(X_z, Y)$, where $Y$ is an $f$-ring and $\mathscr{A}$ is closed under finite intersections, and $X_z$ is a topological space induced by $(X, \mathscr{A})$.  https://cgasa.sbu.ac.ir/article_107064_241c2c0d8f63e5a7a8b455939347fff2.pdf