# Generalizations of Z-supercontinuous functions and Dδ-supercontinuous functions

## DOI:

https://doi.org/10.4995/agt.2008.1804## Keywords:

(almost) z-supercontinuous function, (almost) Dδ-supercontinuous function, (almost) strongly θ-continuous function, Almost continuous function, δ-continuous function, faintly continuous function, uθ-closed graph, θ-closed graph, uθ-limit point, θ-limit po## Abstract

Two new classes of functions, called ‘almost z-supercontinuous functions’ and ’almost D_{δ}-supercontinuous functions’ are introduced. The class of almost z-supercontinuous functions properly includes the class of z-supercontinuous functions (Indian J. Pure Appl. Math. 33(7), (2002), 1097-1108) as well as the class of almost clopen maps due to Ekici (Acta. Math. Hungar. 107(3), (2005), 193-206) and is properly contained in the class of almost D_{δ}-supercontinuous functions which in turn constitutes a proper subclass of the class of almost strongly θ-continuous functions due to Noiri and Kang (Indian J. Pure Appl. Math. 15(1), (1984), 1-8) and which in its turn include all δ-continuous functions of Noiri (J. Korean Math. Soc. 16 (1980), 161-166). Characterizations and basic properties of almost z-supercontinuous functions and almost D_{δ}-supercontinuous functions are discussed and their place in the hierarchy of variants of continuity is elaborated. Moreover, properties of almost strongly θ-continuous functions are investigated and sufficient conditions for almost strongly θ-continuous functions to have u θ-closed (θ-closed) graph are formulated.

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*Appl. Gen. Topol.*, vol. 9, no. 2, pp. 239–251, Oct. 2008.

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