The *I*-density topology is a generalization of the ordinary density
topology to the setting of category instead of measure. This work involves
functions which are continuous when combinations of the *I*-density,
deep-*I*-density, density and ordinary topology are used on the domain and
range. In the process of examining these functions, the *I*-density and
deep-*I*-density topologies are deeply explored and the properties of these
function classes as semigroups are considered.

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**Last modified December 8, 2000.**