We survey a few classes of analytic functions on the disk that have real boundary values almost everywhere on the unit circle. We explore some of their properties, various decompositions, and some connections these functions make to operator theory.

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...

A function f is continuous iff the pre-image f[V ] of any open set V is open again. Dual to this topological property, f is called open iff the image f [U ] of any open set U is open again. Several classical Open Mapping Theorems in Analysis provide a variety of sufficient conditions for openness. By the Main Theorem of Recursive Analysis, computable real functions are necessarily continuous. I...