Sequential Computability of a Function. Effective Fine Space and Limiting Recursion

We consider real sequences in I = [0, 1) and real functions on I. It is first shown that, as for real sequences from I, R-computability (computability with respect to the Euclidean topology) implies “ weak Fine-computability.” Using this result, we show that “ Fine­sequential computability” and “ -sequential computability” are equivalent for effectively locally Fine-continuous functions as well as for Fine-continuous functions