A Galois connection between Turing jumps and limits

Limit computable functions can be characterized by Turing jumps on the inputside or limits on the output side. As a monad of this pair of adjointoperations we obtain a problem that characterizes the low functions and duallyto this another problem that characterizes the functions that are computablerelative to the halting problem. Correspondingly, these two classes are thelargest classes of functions that can be pre or post composed to limitcomputable functions without leaving the class of limit computable functions.We transfer these observations to the lattice of represented spaces where itleads to a formal Galois connection. We also formulate a version of this resultfor computable metric spaces. Limit computability and computability relative tothe halting problem are notions that coincide for points and sequences, buteven restricted to continuous functions the former class is strictly largerthan the latter. On computable metric spaces we can characterize the functionsthat are computable relative to the halting problem as those functions that arelimit computable with a modulus of continuity that is computable relative tothe halting problem. As a consequence of this result we obtain, for instance,that Lipschitz continuous functions that are limit computable are automaticallycomputable relative to the halting problem. We also discuss 1-generic points asthe canonical points of continuity of limit computable functions, and we provethat restricted to these points limit computable functions are computablerelative to the halting problem. Finally, we demonstrate how these results canbe applied in computable analysis