The notion of faintly -continuous function has been introduced. Relationship between this new class of function with similar types of functions has been given. Some characterizations and properties of such function are also being discussed. 1. Introduction In topology, weak forms of open sets play an important role in the generalization of different forms of continuity. Using different forms of open sets, many authors have introduced and studied various types of continuity. In this paper, a unified version of some types of continuity has been introduced from a generalized topological space to a topological space. Generalized topology was first introduced by Császár (see [1–5]). We recall some notions defined in [1]. Let be a nonempty set, and denotes the power set of . We call a class a generalized topology [1], (GT) if and union of elements of belongs to . A set , with a GT on it, is said to be a generalized topological space (GTS) and is denoted by . Let be a topological space. The -closure [6] of a subset of a topological space is defined by , where a subset is called regular open if . A subset of a topological space is called semiopen [7] (resp., preopen [8], -open [9], -open [10], -open [11], -preopen [12], -semiopen [13], and -open [14]) if (resp., , , , , , , and ). A point is in (resp., ) if for each semiopen (resp., preopen) set containing , . A point is called a -cluster [6] (resp., semi- -cluster [15], -cluster [16]) point of denoted by (resp., , if (resp., , ) for every open (resp., semiopen, preopen) set containing . A subset is called -closed (resp., semi- -closed, -closed) if (resp., , - ). The complement of a -closed (resp., semi- -closed, -closed) set is called -open (resp., semi- -open, -open). The family of all -open sets in a topological space forms a topology which is weaker than the original topology. The finite union of regular open sets is said to be -open [17]. A subset of a topological space is said to be -closed [17] if whenever and is -open. A subset of is called -open [18] if for each there exists an open set containing such that . The family of all -open subsets of a space forms a topology on finer than . For any topological space , the collection of all semiopen (resp., preopen, -open, -open, -open, -preopen, -semiopen, -open, -open, semi- -open, -open, -open, and -open) sets are denoted by (resp., , , , , , , , , , ,？ , and ). We note that each of these collections forms a GT on . For a GTS , the elements of are called -open sets and the complement of -open sets are called -closed sets. For , we denote by the intersection
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