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

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