The functions of Computable Analysis are defined by enhancing the capacities of normal Turing Machines to deal with real number inputs. We consider characterizations of these functions using function algebras, known as Real Recursive Functions. I will discuss the basic background, then discuss some recent characterizations of Computable Analysis, and then discuss our new characterization using a simpler function algebra. Our proof is quite different from those of others, using our earlier general techniques: Rather than directly showing that the function algebra can simulate a Turing Machine, we relate the two classes of functions using a series of intermediate classes of functions, via our notion of "approximation". (Joint work with Manuel L. Campagnolo)