Computability of discontinuous functions Functional space approach A gerenal report at Dagstuhl Seminar

We are to discuss how to view notions of computability for discontinuous functions. We con ne ourselves to real-valued functions from some spaces. Our standpoint in studying computability problems in mathematics is doing mathematics. That is, we would like to talk about computable functions and other mathematical objects just as one talks about continuous functions, integrable functions, etc. In any naive notion of computability of a function (on a compact set), uniform continuity is inherent. On the other hand, one often approximates discontinuous functions, for instance in numerical computations and drawing graphs. Very often such approximations are successful and satisfactory. It is therefore meaningful and important to speculate on computability of discontinuous functions. According to our standpoint, it is a mathematical investigation to formulate computability of discontinuous functions and to nd out how it is related to some existing mathematical notions. One possible method of such investigations is to work in some theories of abstract spaces; for example, functional spaces such as Banach spaces and Frechet spaces, metric spaces, uniform topological spaces. The reason why such a method is e ective for the purpose is that an abstract theory of spaces supplies us with a logical framework for mathematics. A functional space is de ned axiomatically, and the objects which satisfy such axioms are those that are well-controlled.