Quasi-continuity (in the sense of Kempisty) generalizes directional continuity of complex-valued functions on open subsets of ℝ n or ℂ n, and in particular provides certain approach regions at every point. We show that these can be used as a proof tool for proving several properties forLebesgue measurable, locally bounded, quasi-continuous functions e.g. that for such a function f the polynomial ring C(M,K)[f] (where K = ℝ or ℂ) satisfies that the equivalence classes under identification a.e. have cardinality one and an asymptotic maximum principle.