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Given an arbitrary real function f , the set D_f of all points where f admits approximate limit is the maximal (with respect to the relation of inclusion except for a nullset) measurable subset of the real line having the properties that the restriction of f to D_f is measurable, and f is approximately continuous at almost every point of D_f . These results extend the well-known fact that a function is measurable if and only if it is approximately continuous almost everywhere. In addition, there exists a maximal G_δ -set C_f (which can be actually constructed from f ) such that it is possible to find a functiong = f almost everywhere, whose set of points of continuity is exactly C_f .