Re: -- Rational -> Rational , real to real and f(f(x)) = x

robert israel wrote :

tommy1729 <tommy1729@xxxxxxxxx> writes:

hi

im looking for functions that satisfy all of the
below :

f maps all rationals to rationals

f maps all reals to reals

and f( f(x) ) = x for real x. (or better all
complex z )

If by saying "complex z" you're hinting that f should
be an entire function,
the only entire functions that are one-to-one are
polynomials of degree 1,

one-to-one is not required.

analytic solutions are preferred.

which leads to rather boring solutions.

yes , of course degree 1 polynomials where not the answers i was hoping for.

i should have stated that i guess.

On the other
hand, there are
nontrivial solutions that are analytic in a
neighbourhood of the real line.

thats intresting by itself.

but i prefer everywhere analytic.

thanks for your comment.


--
Robert Israel
israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics
http://www.math.ubc.ca/~israel
University of British Columbia Vancouver,
BC, Canada

regards
tommy1729
.

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