Is there a bijection $\phi: \mathbb Q \to \mathbb Q$ such that
- $\phi$ is nonlinear: different from $ax+b$,
- $\phi$ is smooth: the extension $\hat{\phi}$ of $\phi$ over $\mathbb R$ is $\mathcal C^2$ ?
What if we require $\mathcal C^\infty$?
$\begingroup$
$\endgroup$
6
-
6$\begingroup$ mathoverflow.net/questions/48910/… $\endgroup$– zntCommented Mar 22, 2016 at 22:11
-
$\begingroup$ $x \mapsto {x^3 \over 1+x^2}$ $\endgroup$– Martin HairerCommented Mar 22, 2016 at 22:16
-
1$\begingroup$ @MartinHairer That doesn't look surjective to me (no rational value of $x$ maps to $1$). $\endgroup$– Todd TrimbleCommented Mar 22, 2016 at 23:16
-
$\begingroup$ As written, this is a special-case of the question znt links to. $\endgroup$– Ryan BudneyCommented Mar 22, 2016 at 23:37
-
$\begingroup$ Sort of, except this question imposes bijectivity. Fortunately, this condition was treated by Pietro Majer in his solution over there. $\endgroup$– Todd TrimbleCommented Mar 23, 2016 at 0:36
|
Show 1 more comment