# Limit computable maximum

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

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> 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?
>
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