This paper aims to present the notion of (i, j)*-fuzzy γ-open set in a fuzzy bitopological space as a parallel form of (i, j)-fuzzy γ-open set due to Tripathy and Debnath (2013) [17] and show that both of them are independent concepts. Then we extend our study to (i, j)*-generalized fuzzy γ-closed set and (i, j)*-γ-generalized fuzzy closed set. We show that (i, j)*-γ-generalized fuzzy closed set and (i, j)*-generalized fuzzy γ-closed set are also independent of each other in nature. Though every (i, j)*-fuzzy γ-closed set is a (i, j)*-generalized fuzzy γ-closed set but with (i, j)*-γ-generalized fuzzy closed set, the same relation is not linear. Similarly though every (i, j)*-fuzzy closed set is (i, j)*-γ-generalized fuzzy closed set but it is independent to (i, j)*-generalized fuzzy γ-closed set. Various properties related to (i, j)*-generalized fuzzy γ-closed set are also studied. Finally, (i, j)*-generalized fuzzy γ-continuous function and (i, j)*-generalized fuzzy γ-irresolute functions are introduced and interrelationships among them are established. We characterized these functions in different directions for different applications.This paper aims to present the notion of  (i,j)* fuzzy γ-open set in a fuzzy bitopological space as a parallel form of  (i,j) fuzzy γ-open set due to Tripathy and Debnath (2013)  [Tripathy, B. C., & Debnath, S. (2013),   γ-Open sets and γ-Continuous Mappings in Fuzzy Bitopological Spaces, Journal of Intelligence and Fuzzy Systems, 24, 631-635] and show that both of them are independent concepts. Then we extend our study to (i,j)* generalized fuzzy γ-closed set and (i,j)* γ-generalized fuzzy closed set. We show that (i,j)* γ-generalized fuzzy closed set and (i,j)* generalized fuzzy γ-closed set are also independent of each other in nature. Though every (i,j)* fuzzy γ-closed is a (i,j)* generalized fuzzy γ-closed set but with (i,j)* γ-generalized fuzzy closed set, the same relation is not linear. Similarly though every (i,j)* fuzzy closed set is (i,j)* γ-generalized fuzzy closed set but it is independent to (i,j)* generalized fuzzy γ-closed set. Various properties related to (i,j)* generalized fuzzy γ-closed set are also studied. Finally, (i,j)* generalized fuzzy γ-continuous function and (i,j)* generalized fuzzy γ-irresolute functions are introduced and interrelationships among them are established. We characterized these functions in different directions for different application.