Abstract. A new strong variant of continuity called 'i?-supercontinuity' is introduced. Basic properties of R-supercontinuous functions are studied and their place in the hierarchy of strong variants of continuity that already exist in the literature is elaborated. It is shown that fl-supercontinuity is preserved under the restriction, shrinking and expansion of range, composition of functions, products and the passage to graph function. The class of .R-supercontinuous functions properly contains each of the classes of (i) strongly 0-continuous functions introduced by Noiri and also studied by Long and Herrington; (ii) D-supercontinuous functions; and (iii) F-supercontinuous functions; and so include all z-supercontinuous functions and hence all clopen maps (= cl-supercontinuous functions) introduced by Reilly and Vamnamurthy, perfectly continuous functions defined by Noiri and strongly continuous functions due to Levine. Moreover, the notion of r-quotient topology is introduced and its interrelations with the usual quotient topology and other variants of quotient topology in the literature are discussed. Retopologization of the domain of a function satisfying a strong variant of continuity is considered and interrelations among various coarser topologies so obtained are observed.

A strong variant of continuity called 'F -supercontinuity' is introduced. The class of F -supercontinuous functions strictly contains the class of z-supercontinuous functions (Indian J. Pure Appl. Math. 33 (7) (2002), 1097-1108) which in turn properly contains the class of cl-supercontinuous functions (≡ clopen maps) (Appl. Gen. Topology 8 (2) (2007), 293-300; Indian J. Pure Appl. Math. 14 (6) (1983), 762-772). Further, the class of F -supercontinuous functions is properly contained in the class of R-supercontinuous functions which in turn is strictly contained in the class of continuous functions. Basic properties of F -supercontinuous functions are studied and their place in the hierarchy of strong variants of continuity, which already exist in the mathematical literature, is elaborated. If either domain or range is a functionally regular space (

Abstract. Two new weak variants of continuity called 'R-continuity'and 'F-continuity' are introduced. Their basic properties are studied and their place in the hierarchy of weak variants of continuity, that already exist in the literature, is elaborated. The class of R-continuous functions properly contains the class of continuous functions and is strictly contained in each of the three classes of (1) faintly continuous functions studied by Long and Herrignton (Kyungpook Math. J.

In this paper, closedness of certain classes of functions in

A new class of functions called "Rz-supercontinuous functions" is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity that already exist in the literature is elaborated. The class of Rz-supercontinuous functions properly includes the class of R cl-supercontinuous functions, Tyagi, Kohli, Singh (2013), which in its turn contains the class of cl-supercontinuous (≡ clopen continuous)