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Post Closed as "Duplicate" by Gerry Myerson, Ryan Budney, Alexandre Eremenko, Alexey Ustinov, Alex Degtyarev
deleted 1 character in body; edited title
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Marco Disce
Marco Disce
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Nonlinear regularsmooth bijection from $\mathbb Q$ to itself

Is there a bijection $\phi: \mathbb Q \to \mathbb Q$ such that

  • $\phi$ is nonlinear: different from $ax+b$,
  • $\phi$ is regularsmooth: the extension $\hat{\phi}$ of $\phi$ over $\mathbb R$ is $\mathcal C^2$ ?

What if we require $\mathcal C^\infty$?

Nonlinear regular bijection from $\mathbb Q$ to itself

Is there a bijection $\phi: \mathbb Q \to \mathbb Q$ such that

  • $\phi$ is nonlinear: different from $ax+b$,
  • $\phi$ is regular: the extension $\hat{\phi}$ of $\phi$ over $\mathbb R$ is $\mathcal C^2$ ?

What if we require $\mathcal C^\infty$?

Nonlinear smooth bijection from $\mathbb Q$ to itself

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 $\mathcal C^\infty$?

Source Link
Marco Disce
Marco Disce
  • 303
  • 3
  • 4
  • 8

Nonlinear regular bijection from $\mathbb Q$ to itself

Is there a bijection $\phi: \mathbb Q \to \mathbb Q$ such that

  • $\phi$ is nonlinear: different from $ax+b$,
  • $\phi$ is regular: the extension $\hat{\phi}$ of $\phi$ over $\mathbb R$ is $\mathcal C^2$ ?

What if we require $\mathcal C^\infty$?