This is a truly great paper, and I am glad to have read it. But there is something wrong with the theorem at the beginning of 4.14 ("all computable functions that are defined for all computable real arguments and return computable real results are continuous"), probably that the claim is over-generalized (or maybe just that I am a high-school mathematician). The domain of the signum function obviously includes the computable reals. The codomain is {0, 1, -1}, which are all computable. Yet signum is not continuous at 0. This can be generalized to other stepwise functions with any countable number of discontinuous points. What gives?

On Tue, Feb 1, 2022 at 12:15 PM Bradley Lucier <lucier@math.purdue.edu> wrote:

I wrote a paper on the computable reals with a page of (Gambit) Scheme
code at the end that implements some of the examples. The code was
organized for clarity of presentation (base 10, easiest algorithms,
etc.), so it's not production quality, but there are references for
people who are interested:

https://scholarship.claremont.edu/jhm/vol12/iss1/25/

I made sure that the typesetting didn't screw up the code, that you can
just copy it to a Gambit REPL and everything works.

And Section 3 tells a funny story about me learning how to compute
square roots by hand. (Perhaps only a mathematician could write that
sentence!)

Brad

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