Many studies have been conducted on properties of bitopological spaces and aspects of continuity over a long period of time and different results have been obtained so far. However, pointwise characterization of various aspects of continuity has not been done in bitopological spaces. Moreover, our work is aiming at establishing particular separation criteria for bitopological and spaces where N > 2. This therefore calls for an in depth study of continuity and separability in bitopological spaces. The objectives of the study were to: characterize notion of ij-continuity in bitopological spaces; establish separation criteria for bitopological spaces via ij-continuity; and determine extensions of continuity and separability in N-topological spaces. The methodologies involved use of criterion for continuity, criteria for inverse continuity, separation axioms and conditions for normality. The results showed that various continuity notions such as p_, θ_ and pd exist in bitopological spaces. For separation criteria, the results showed that if bitopological spaces are T0, T1, T2 and T52 properties are both topological and hereditary. For extension and separability in N-topological spaces results indicated that properties can be naturally extended to N-topological spaces. The results obtained are useful in studying topological deformations such as stretching which is fundamental in understanding the shape and structure of the universe and formulations of real functions and topological mappings. Our results also help in deep understanding of molecular biology more particularly on DNA structure. Our results also play a great role in understanding the applications of computer topology such as line, ring, star and hybrid topologies.