Let X be a topological space and (Y,d) be a metric space. If f: X → Y is a function then there is a function af: X → [0, ∞] such that f is almost continuous at x if and only if af (x) = 0. Some properties of this function are investigated.
The concepts of I-R closed set, AI-R-set, αIM1-set, αIM2-set, αIN1-set, αIN2-set, αIN3-set, αIN4-set and αIN5-set are introduced via idealization. New decompositions of some weaker forms of continuity are obtained by using these sets.