What about the computable reals? Our computer programs are, in a sense, capable of emitting the digits of any computable real number, and we have arithmetic (Go

What about the computable reals? Our computer programs are, in a sense, capable of emitting the digits of any computable real number, and we have arithmetic (Gosper continued-fraction arithmetic) on those digits. The fact that values might be discrete does not change that programs might be continuous over their output range; functions from computable reals to computable reals can be continuous.