Density Continuity Versus Continuity

Real-valued functions of a real variable which are continuous with respect to the density topology on both the domain and the range are called density continuous. A typical continuous function is nowhere density continuous. The same is true of a typical homeomorphism of the real line. A subset of the real line is the set of points of discontinuity of a density continuous function if and only if it is a nowhere dense Fσ set. The corresponding characterization for the approximately continuous functions is a Ærst category Fσ set. An alternative proof of that result is given. Density continuous functions belong to the class Baire*1, unlike the approximately continuous functions.