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 - supercontinuous functions. We consider these two classes of functions from the perspective of changes of topologies. Inparticular,weshowthateachofthesevariantsofcontinuitycoincideswiththeclassicalnotion 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.