Computability

For the purposes of this paper, a function 44#44 is computable if it has a computable realizer 45#45. Computability on 47#47 can be defined e.g. via Turing machines with (read-only) input tapes and (write-only) output tapes, in additions to the usual (read-write) working tapes [35,62,64]. In practice, an intuitive understanding of computability on 47#47 is enough for most purposes.

We have seen that computable functions 44#44 are continuous. More generally, a computable partial function is continuous on its domain of definition. For example, the function 1/x is continuous and computable on 48#48, but cannot be extended to a continuous function at 0. In practice, one gets non-termination at points of discontinuity.