In this paper, we introduce and analyse new classes of generalized continuous mappings in topological spaces using newly defined classes of ω-open sets. Specifically, ω-h-continuous and ω-h*-continuous functions are defined and investigated; a new generalized class, ω-h-irresolute functions is also proposed within the same framework. A few fundamental properties and characterizations of these mappings are described. Moreover, the associations among the proposed classes and some previously known types of generalized continuous and irresolute functions are analysed systematically. This study supports some inclusion results between the associated classes of generalized open and closed sets, and employs them to infer corresponding implications between the introduced types of mappings. Additionally, it discusses characterization theorems for ω-h-continuous and ω-h-irresolute functions with respect to inverse images, closures and interiors. Not only that, we research how these functions behave under restriction and composition, and we learn some of the conditions which contribute to preserving the newly introduced continuity properties. Some examples are provided to show that certain converse implications do not in general hold, and to elucidate the independence of elements and sharpness of the reported results. Results derived from that research expand and synthesize a number of the ideas already built in generalized topology, as well as providing a larger framework for examining generalized continuity in topological spaces generally.