Two new classes of functions between topological spaces have been defined recently, and their basic properties have been studied. They are called almost z-supercontinuous functions and almost D-delta-supercontinuous functions. We consider these two classes of functions from the perspective of changes of topologies. In particular, we show that each of these variants of continuity coincides with the classical notion of continuity when the domain and codomain of the function under consideration have been retopologized appropriately. Some of the consequences of this situation are examined in this paper.