Re: i apologize : of course f '' (x) = / = 0.

In article
<a8979d9e-ed9d-4f23-a670-7faad10faba8@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Mate <mmatica@xxxxxxxxxxx> wrote:

On Feb 27, 7:51 pm, quasi <qu...@xxxxxxxx> wrote:
On Wed, 27 Feb 2008 12:33:45 EST, tommy1729 <tommy1...@xxxxxxxxx>
wrote:



On 27 Feb, 16:13, tommy1729 <tommy1...@xxxxxxxxx>
wrote:
tommy1729 <tommy1...@xxxxxxxxx> wrote:
hi all

let f(x) be an analytic function mapping
all
rationals to rationals (
and all reals to reals ).

i apologize : of course f '' (x) = / = 0.
is what i meant.

So try f(x) = x^2
and f''(x) = 2

all rationals to all rationals.

i forgot the second 'all'.

thus f must map all rationals to rationals

and the inverse of f must map all rationals to rationals

EQUALS f maps all rationals to all rationals.

hèhè

sorry

So to state a complete problem, I think your question is this ...

Does there exist a twice differentiable function f : R --> R such that

(1) f(x) is rational iff x is rational.

(2) f''(x) is nonzero for all x in R.

?


Yes, even an analytic f. See the sci.math post (1995!):

http://groups.google.com/group/sci.math/browse thread/thread/8bc932776f3648a9

This URL didn't work for me, but it did contain enough information
for me to find the thread. The subject was "Functions preserving
rationality?"

--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.