On Some Computability Notions for Real Functions

A widely used approach to computability of real functions is the one in Grzegorczyk’s style originating from [1]. This approach uses computable transformations of infinitistic names of real numbers, as well as general quantifiers over these names. Other approaches allow avoiding the use of such names at least in some cases. An approach of this other kind is, for instance, the one of Tent and Ziegler from [5]. In the present paper, the equivalence of a certain approach in Grzegorczyk’s spirit and one in the spirit of Tent and Ziegler is shown under some general assumptions.