MAGIC SETS

In this paper we study magic sets for certain families H subset of R-R which are subsets M subset of R such that for all functions f, g is an element of H we have that g[M] subset of f [M] double right arrow f = g. Specifically we are interested in magic sets for the family g of all continuous functions that are not constant on any open subset of R. We will show that these magic sets are stable in the following sense: Adding and removing a countable set does not destroy the property of being a magic set. Moreover, if the union of less than c meager sets is still meager, we can also add and remove sets of cardinality less than c without destroying the magic set. Then we will enlarge the family G to a family F by replacing the continuity with symmetry and assuming that the functions are locally bounded. A function f : R -> R is symmetric iff for every x is an element of R we have that lim(h down arrow 0) 1/2 (f (x + h) + f (x - h)) = f (x). For this family of functions we will construct 2(c) pairwise different magic sets which cannot be destroyed by adding and removing a set of cardinality less than c. We will see that under the continuum hypothesis magic sets and these more stable magic sets for the family F are the same. We shall also see that the assumption of local boundedness cannot be omitted. Finally, we will prove that for the existence of a magic set for the family F it is sufficient to assume that the union of less than c meager sets is still meager. So for example Martin's axiom for sigma-centered partial orders implies the existence of a magic set.