  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.
