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 “Fine-sequential computability” and “L∗-sequential computability” are equivalent for effectively locally Fine-continuous functions as well as for Finecontinuous functions.