The *I*-density topology T_{I} 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
C_{I} of ordinary continuous functions
f:[0,1]-->**R**
which have at least one point of
*I*-density continuity is a first category subset of

equipped with the uniform norm. It is also proved, that the class of

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**Last modified May 1, 1999.**