  Algebraic characterizations of the computational aspects of functions defined
over the real numbers provide very effective tool to understand what
computability and complexity over the reals, and generally over continuous
spaces, mean. This is relevant for both communities of computer scientists and
mathematical analysts, particularly the latter who do not understand (and/or
like) the language of machines and string encodings. Recursive analysis can be
considered the most standard framework of computation over continuous spaces,
it is however defined in a very machine specific way which does not leave much
to intuitiveness. Recently several characterizations, in the form of function
algebras, of recursively computable functions and some sub-recursive classes
were introduced. These characterizations shed light on the hidden behavior of
recursive analysis as they convert complex computational operations on
sequences of real objects to "simple" intuitive mathematical operations such as
integration or taking limits. The authors previously presented a framework for
obtaining algebraic characterizations at the complexity level over compact
domains. The current paper presents a comprehensive extension to that
framework. Though we focus our attention in this paper on functions defined
over the whole real line, the framework, and accordingly the obtained results,
can be easily extended to functions defined over arbitrary domains.
