Set of continuity points of a real function
Motivated by the examples of the Dirichlet's no-where continuous function and Thomae's function, we know that a function can have discontinuities at the reals, or at the rationals. My question is whether the set of discontinuities can actually be at the irrationals only. Froda's theorem tells us that such a function cannot be monotone, so that cuts down on the search. Are there other obstructions? Can we exhibit such a function?