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.