This paper is concerned with four topologies on the
real line: the ordinary topology, density topology, *I*-density topology and
deep-*I*-density topology. Using different topologies on the domain and range,
these four topologies determine sixteen different ways a real function can be
continuous. These continuities are examined and their relationships are
determined. In addition, they are compared with the classes of analytic,
C^{\infty}, Baire*1, and Baire~1 functions.

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**Last modified April 30, 1999.**