I don't think med(.,.) is continuous as a map from QxQ to Q (which rules out the possibility of a continuous extension to RxR). Specifically, I claim that on every neighborhood of (1,2) in QxQ, med(p/q,r/s) takes on values arbitrarily close to 1 (when p,q are much larger than r,s) and values arbitrarily close to 2 (when r,s are much larger than p,q). But maybe the moral here is that we should use a different topology on QxQ. Jim On Friday, February 3, 2017, Dan Asimov <asimov@msri.org> wrote:
A kind soul has pointed out that I spaced out when typing the mediant.
For positive fractions p/q and r/s assumed to be in lowest terms, their mediant (CORRECTED) is
med(p/q, r/s) = (p+r)/(q+s)
. (It's well known that med(p/q, r/s) always lies strictly between p/q and r/s.)
—Dan
On Feb 3, 2017, at 1:08 AM, Dan Asimov <asimov@msri.org <javascript:;>> wrote:
The mediant of any 2 positive rational numbers p/q and r/s is
[WRONG:] med(p/q, r/s) = (p+q)/(r+s).
This can be thought of as a mapping
med: Q+ x Q+ —> R
from the cartesian product of the positive rationals with itself to the reals.
Question: --------- Is med the restriction to Q+ x Q+ of some continuous function
Med: R x R —> R
???
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