A smooth function which is nowhere real analytic, and preserves rationality of its argument

Question

There are examples $\!^{[1]}$$\!^{[2]}$ of continuous infinitely differentiable (class $C^\infty$) functions $\mathbb R\to\mathbb R$ that are nowhere real analytic. I wonder if it is possible to construct such an example that preserves rationality of its argument, i.e. $f(x)\in\mathbb Q$ iff $x\in\mathbb Q$.

Answer 1

Values of Fabius function or it's first prime in dyadic rationals are rational,
for example.

Concernig non dyadic rational it seems to be an open question indeed.