Beyond Recursive Real Functions

All recursive real functions are continuous; in fact all theB-recursive real functions are continuous for any oracleB, simply because Turing machines computing them are finite objects. But simple discontinuous functions such as step functions have to be in some sense “easy” if they have recursive values and break points, although they are not computable by the usual definition. It seems unfair to label them noncomputable in the entire region just because of a few break points. In this paper, we investigate the properties of broader classes ofalmost everywhere recursive, weakly almost everywhere recursive, andrecursively approximablereal-valued functions, which capture these “easy” step functions and many other nonrecursive functions. Recursive versions of the classical Lusin and Egoroff Theorems are proved. We also characterize the property of the limit of a recursive sequence of functions and show that different notions of convergence (uniform, pointwise, or in measure) will result in different characterizations of the limiting function.

Authors

• Chun-Kuen Ho

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