Recursion Theory on the Reals and Continuous-Time Computation

We define a case of recursive functions on the reals analogous to the classical recursive functions on the natural numbers, corresponding to a conceptual analog computer that operates in continuous time. This class turns out to be surprisingly large, and includes many functions which are uncomputable in the traditional sense. We stratify this class of functions into a hierarchy, according to the number of uses of the zero-finding operator $\mu$. At the lowest level are continuous functions that are differentially algebraic, and computable by Shannon's General Purpose Analog Computer. At higher levels are increasingly discontinuous and complex functions. We relate this $\mu$-hierarchy to the Arithmetical and Analytical Hierarchies of classical recursion theory.