Beyond Recursive Real Functions

All the recursive real functions are continuous; in fact all the B-recursive real functions are continuous for any oracle B, simply because Turing Machines computing them are nite objects. But simple functions like step functions have to be in some sense be \easy" if they have recursive values and break points, although by our usual deeni-tion, they are not computable at all. So it seems unfair to label them not computable in the entire region just because of a few break points. In this paper, we investigate the properties of broader classes of almost everywhere recursive, weakly almost everywhere recursive and recursively approximable real-valued functions, which captures these \easy" step functions and many other nonrecursive functions. A recursive version of the classical Lusin's and Egoroo's theorem are proved and we also try to characterize the property of the limit of a recursive sequence of functions and show that diierent notion of convergence (uniform, pointwise or in measure) will result in diierent characterization of the limiting function. A recursive real-valued function on a bounded domain is recursive if there is an eeective way (say by a Turing machine) to approximate its value at a point x as accurate as we like, if an approximation to x itself is given (say, as an oracle). In the following, let N be the set of nonnegative integers; D be the set of dyadic rationals; and D n be the set of dyadic rationals of the form m 2 n where m is any integer and n is a positive integer, and so we have D = S n D n. Let CF x (Cauchy functions for x) denote the set of all dyadic rational functions such that (n) 2 D n and j(n) ? xj 2 ?n. There is one special x 2 CF x , called the standard Cauchy function for x which has the property that j x (n)j x < j x (n)j + 2 ?n .