Irrational-Based Computability of Functions

We investigate a sort of a unifying theory of computability of real functions, continuous or discontinuous, called here "irrational-based" (IB-) computability. All the examples which are computable in our various theories presented previously are IB-computable. The basic requirements of computability, the sequential computability and the effective continuity, are defined relative to computable irrational real sequences. The family of IB-computable functions is closed under IB-effective convergence. In order to certify the fruitfulness of IB-computability, quite a number of examples are presented.