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