Is there a bijection $\phi\colon \mathbb Q \to \mathbb Q$ such that
- $\phi$ is nonlinear (i.e., different from $x\mapsto ax+b$),
- $\phi$ is regular: the extension $\hat{\phi}$ of $\phi$ over $\mathbb R$ is $\mathcal C^2$?
What if we require $\mathcal C^\infty$?