Computability of a function with jumps: Effective uniformity and limiting recursion

Given a strictly increasing computable sequence (called a base sequence) of real numbers (with respect to the Euclidean topology), one can induce an effective uniformity for the real line, where the elements in the base sequence are regarded as isolated. The relation between two extended notions of computability of real sequences, one with respect to the Euclidean space with a limiting recursive modulus of convergence and one with respect to the induced uniform space, is discussed. As a consequence, we prove the equivalence of two extended notions of sequential computability (called L- and A- sequential computability) of a real function. This indicates that the two extended notions of sequential computability provide computational mechanisms of the same power. We will then characterize a piecewise continuous function to be computable (called para-computable here) as being L- (hence A-) sequentially computable and piecewise effectively continuous.