In the answer to this close question I remarked how to make an entire, non polynomial function bijecting two assigned countable dense sets $A$ and $B$.

Also, you can make a non-analytic $C^\infty$ self-diffeo of $\mathbb{R}$ that bijects $A$ and $B$ with similar procedure. Say a $C^\infty$ diffeo, which is not the identity map, but with a tangency of infinite order to the identity map at $0$.

In the answer to this close question I remarked how to make an entire, non polynomial function bijecting two assigned countable dense sets $A$ and $B$.

Also, you can make a non-analytic $C^\infty$ self-diffeo of $\mathbb{R}$ that bijects $A$ and $B$ with similar procedure. Say a $C^\infty$ diffeo, which is not the identity map, but with a tangency of infinite order to the identity map at $0$.

In the answer to this close question I remarked how to make an entire, non polynomial function bijecting two assigned countable dense sets $A$ and $B$.

Also, I'm pretty sure you can make a non-analytic $C^\infty$ self-diffeo of $\mathbb{R}$ that bijects $A$ and $B$ with similar procedure. Say a $C^\infty$ diffeo, which is not the identity map, but with a tangency of infinite order to the identity map at $0$.

In the answer to this close question I remarked how to make an entire, non polynomial function bijecting two assigned countable dense sets $A$ and $B$.

Also, you can make a non-analytic $C^\infty$ self-diffeo of $\mathbb{R}$ that bijects $A$ and $B$ with similar procedure. Say a $C^\infty$ diffeo, which is not the identity map, but with a tangency of infinite order to the identity map at $0$.