In computable analysis, sequences of rational numbers which effectively converge to a real number x are used as the (rho-) names of x. A real number x is computable if it has a computable name, and a real function f is computable if there is a Turing machine M which computes f in the sense that, M accepts any rho-name of x as input and outputs a rho-name of f(x) for any x in the domain of f. By weakening the effectiveness requirement of the convergence and classifying the converging speeds of rational sequences, several interesting classes of real numbers of weak computability have been introduced in literature, e.g., in addition to the class of computable real numbers (EC), we have the classes of semi-computable (SC), weakly computable (WC), divergence bounded computable (DBC) and computably approximable real numbers (CA). In this paper, we are interested in the weak computability of continuous real functions and try to introduce an analogous classification of weakly computable real functions. We present definitions of these functions by Turing machines as well as by sequences of rational polygons and prove these two definitions are not equivalent. Furthermore, we explore the properties of these functions, and among others, show their closure properties under arithmetic operations and composition.

The Turing degree of a real number x is defined as the Turing degree of its binary expansion. This definition is quite natural and robust. In this paper we discuss some basic degree properties of semi‐computable and weakly computable real numbers introduced by Weihrauch and Zheng. We show that there are two real numbers of c.e. binary expansions such that their difference does not have an ω.c.e. Turing degree.

We compare the weak completeness notions for E in the sense of Lutz's resource-bounded measure theory (1992) with respect to the standard polynomial time reducibilities. Our results parallel results for classical completeness by Watanabe (1987) and others. We show that the weak completeness notions for 1-query reductions coincide: A set is weakly complete for E under 1-truth-table reducibility iff it is weakly complete for length-increasing one-one reducibility. For most of the other polynomial reducibilities, however, we obtain separations of the weak completeness notions where these reducibilities differ on E (Ladner et al. (1975)). In fact our separations simultaneously hold for the corresponding weak completeness notions for E and E/sub 2/, for the classical completeness notions, and for the weak completeness notions in the sense of the resource-bounded Baire category concepts of Ambos-Spies et al. (1988) and Ambos-Spies (1995).