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 F_{\sigma}
set. The corresponding characterization for the
approximately continuous functions is a first category
F_{\sigma} set. An alternative proof of that result is given.
Density continuous functions belong
to the class Baire*1, unlike the approximately continuous functions.

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**Last modified December 18, 2000.**