Limit computable maximum

James Moody james.moody at
Mon Aug 16 13:49:53 EDT 2021

> I'm assuming you are using "computable function" here to mean f is given
by a Turing functional which takes a fast Cauchy sequence of rationals
converging to a real number x in [0,1] and returns a fast Cauchy sequence
of rational numbers converging to f(x).

In that case, [0,1] is compact, so f is not only continuous but uniformly
continuous. The maximum of f over [0,1] is thus equal to the limit in k of
the maximum of f over the rationals in [0,1] with denominator 2^k. The
elements of the of this sequence are uniformly computable real numbers. We
can use these to create a monotonically non-decreasing sequence of
rationals converging to the maximum of f.

Is there a computable function f : [0, 1] --> R that has no limit

computable maximum?
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20210816/ee00275a/attachment-0001.html>

More information about the FOM mailing list