Sequential Computability of a Function: Diagonal Space and Limiting Recursion

We consider the real sequences in I=[0,1) and real functions on I. A computability notion with respect to the uniformity {Un}, where Un(x)=[k2n,k+12n) if x∈[k2n,k+12n) will be called D-computability. An R-computable sequence from I will be shown to be approximated by a recursive sequence of rational numbers with a limiting recursive modulus of convergence with respect to {Un}. Using this result, we relate two extended notions of sequential computability of a function or a function sequence, one formulated in terms of limiting recursion and one in terms of {Un}.