Computable Real Functions : Type 1

Based on the Turing machine model there are essentially two diierent notions of computable functions over the real numbers. The eeective functions are deened only on computable real numbers and are Type 1 computable with respect to a numbering of the computable real numbers. The eeectively continuous functions may be deened on arbitrary real nunbers. They are exactly those functions which are Type 2 computable with respect to an appropriate representation of the real numbers. We characterize the Type 2 computable functions on computable real numbers as exactly those Type 1 computable functions which satisfy a certain additional condition concerning their domain of deenition. Our result is a sharp strengthening of the well-known continuity result of Tseitin and Moschovakis for eeective functions. The result is presented for arbitrary computable metric spaces.