Decomposition of fuzzy continuity and fuzzy ideal continuity via fuzzy idealization

Recently, El-Naschie has shown that the notion of fuzzy topology may be relevant to quantum paretical physics in connection with string theory and E-infinity   space time theory. In this paper, we study the concepts of r-fuzzy semi-I-open, r-fuzzy pre-I-open, r-fuzzy αα-I-open and r-fuzzy ββ-I-open sets, which is properly placed between r-fuzzy openness and r-fuzzy αα-I-openness (r-fuzzy pre-I-openness) sets regardless the fuzzy ideal topological space in Ŝostak sense. Moreover, we give a decomposition of fuzzy continuity, fuzzy ideal continuity and fuzzy ideal αα-continuity, and obtain several characterization and some properties of these functions. Also, we investigate their relationship with other types of function.