The objective of this paper is to propose a new class of functions called Nano fuzzy continuous functions which can be defined on Nano fuzzy topological spaces and to derive their characterizations in terms of Nano fuzzy closed sets, Nano fuzzy closure and Nano fuzzy interior. The concept of Nano fuzzy topological space with respect to the fuzzy subset  of the universe is used here which is defined in terms of Nano fuzzy lower and Nano fuzzy upper approximations of  Approximation operators are useful to find the relation between rough set theory and topology. In this paper we have tried to fuzzify the continuous functions with crisp approximations. Also, we have tried to extend our work by defining Nano fuzzy open maps, Nano fuzzy closed maps and Nano fuzzy homeomorphism.

Index Terms- Nano fuzzy topological space, Nano fuzzy open set, Nano fuzzy closed set, Nano fuzzy interior, Nano fuzzy closure, Nano fuzzy continuous functions, Nano fuzzy open maps, Nano fuzzy closed maps, Nano fuzzy homeomorphisms.