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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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Abstract. It is shown that all approximate symmetric derivatives of measurable functions are in Baire class one. Further, if/is a measurable function which is finite a.e., then its upper and lower approximate symmetric dérivâtes are in Baire class three.