Category theorems concerning I-density continuous functions

by

Krzysztof Ciesielski & Lee Larson

Fund. Math. 140 (1991), 79-85

The I-density topology TI on R is a refinement of the natural topology. It is a category analogue of the density topology. This paper is concerned with I-density continuous functions; i.e., the real functions that are continuous when the I-density topology is used on the domain and the range. It is shown, that the family CI of ordinary continuous functions f:[0,1]-->R which have at least one point of I-density continuity is a first category subset of

C([0,1])={f:[0,1]-->R: f is continuous}

equipped with the uniform norm. It is also proved, that the class of I-density continuous functions, equipped with the topology of uniform convergence, is of first category in itself. These results remain true when the I-density topology is replaced by the deep I-density topology.

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Last modified May 1, 1999.