Computable invariance

AbstractIn Computable Analysis each computable function is continuous and computably invariant, i.e. it maps computable points to computable points. On the other hand, discontinuity is a sufficient condition for non-computability, but a discontinuous function might still be computably invariant. We investigate algebraic conditions which guarantee that a discontinuous function is sufficiently discontinuous and sufficiently effective such that it is not computably invariant. Our main theorem generalizes the First Main Theorem of Pour-El and Richards (cf. [20]). We apply our theorem to prove that several set-valued operators are not computably invariant