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In this paper, the families of functions continuous with respect to some family of subsets of the real line as a domain are considered. There is proved, among others, that if the family of subsets has some special property, then each function continuous with respect to this family is quasi-continuous ( Corollary 2.2.1 ). We say that two families of subsets of the real line are equivalent if the families of functions which are continuous in generalized sense with respect to these families are equal. Limited considerations to the families which are generalized topologies with additional properties, we find in equivalence classes the smallest and the largest family with respect to inclusion (see  Theorem 3.5 ).

The Smallest and the Largest Families of Some Classes of 𝒜-Continuous Functions