Notions of computability for real functions (and functions betweenPolish spaces) have been studied since the 1950s, using two differentapproaches. The first considers computable real functions on all realnumbers (the variant of this approach via type-2-recursion isessentially equivalent). It is closely related to the representations ofanalytical objects (such as Polish spaces) used in intuitionisticanalysis and Bishop-style constructive analysis. In the second approach,which has been investigated mainly in the Russian school ofconstructivism, computable real functions are defined only on thecomputable reals.

Edalat and Sünderhauf provide an alternative to the firstapproach via an effective theory of continuous domains. This theory goesback to D. Scott, but its applications tocomputable analysis have been pursued mainly by the first author inrecent years, in a series of interesting papers that present a generalprogram of continuous domains in computable analysis. Here, the authorsconsider computable functions on R and C; they refer to a subsequent paper for extensions tometric spaces. They show that their domain-theoretic notions forcomputable real numbers and computable real functions coincide with theclassical ones from the first approach mentioned above. Thedomain-theoretic approach to computability on the real line is certainlyelegant, but it seems to have a weakness as well: the equivalenceresults mentioned above are based on general computability and, as theystand, do not relativize to subrecursive complexity classes. In fact, anobstacle to treating complexity issues in the domain-theoretic frameworkis that a quantitative notion of convergence is missing. In comparisonto this, the classical (first) approach directly yields a complexity theory  for real functions(and functionals of real functions) as developed, for example, byH. Friedman andK.-I. Ko.