On Wed, Feb 10, 2016 at 9:44 AM, Warren D Smith <warren.wds@gmail.com> wrote:
And one can similarly ask: is there an analytic or Cinfinity function that maps algebraic numbers to algebraic numbers, but is not an algebraic function?
Here's another interesting real valued function F(X) of real X. Let [a0; a1,a2,a3,a4...] be the continued fraction expansion of X. For each a[j] which is (i) greater than 4, and (ii) both its neighbors a[j+1] and a[j-1] also are greater than 4, replace a[j] by a[j]+1. This editing process converts X into F(X).
I don't think this is continuous. Let X be a rational number whose continued fraction expansion ends with [...2, 4, 4, 4]. f(X) has a continued fraction expansion that ends with [...2, 4, 5, 4]. But a number very slightly larger or smaller than X (depending on whether X has an even or odd number of convergents) will have a continued fraction expansion that has in this spot [...2, 4, 4, 3, 1, ...], and f maps this to itself.
(5) Is F(X) continuous? (I think yes. Certainly if X and Y are both rational and |X-Y|-->0, then |F(X)-F(Y)|-->0.)
General rule when looking for the flaw in an argument; look for the statement that something is "certainly" or "obviously" true. Andy Latto andy.latto@pobox.com