An Effective Operator, Continuous but Not Partial Recursive

Introduction. It is known that under many conditions, effective operators will be partial recursive, ([MS], [KLS], [L]). On the other hand, certain pathological examples have been constructed by Friedberg [F] and Pour-El [P] to show that effective operators are not always partial recursive. Pour-El has observed that although it is well known that all partial recursive operators are continuous, the effective but not partial recursive operators of [F] and [P] are not continuous, and she has raised the question of the existence of effective operators which are continuous but not partial recursive. It is easy to see that all partial recursive operators are not just continuous, but are in fact "effectively continuous." This enables us to answer Pour-El's question by constructing an effective operator which is continuous but not "effectively continuous." Since it is continuous, our example of an effective but not partial recursive operator is perhaps less pathological than earlier examples.2