A foundation for real recursive function theory

The class of recursive functions over the reals, denoted by REC(R), was introduced by Cristopher Moore in his seminal paper written in 1995. Since then many subsequent investigations brought new results: the class REC(R) was put in relation with the class of functions generated by the General Purpose Analogue Computer of Claude Shannon; classical digital computation was embedded in several ways into the new model of computation; restrictions of REC(R) were proved to represent different classes of recursive functions, e.g., recursive, primitive recursive and elementary functions, and structures such as the Ritchie and the Grzergorczyk hierarchies. The class of real recursive functions was then stratified in a natural way, and REC(R) and the analytic hierarchy were recently recognised as two faces of the same mathematical concept. In this new article, we bring a strong foundational support to the Real Recursive Function Theory, rooted in Mathematical Analysis, in a way that the reader can easily recognise both its intrinsic mathematical beauty and its extreme simplicity. The new paradigm is now robust and smooth enough to be taught. To achieve such a result some concepts had to change and some new results were added.