If a function is discontinuous on $\mathbb Q$, is it necessarily discontinuous on $\mathbb R \setminus \mathbb Q$? Suppose $f : \mathbb{R} \rightarrow \mathbb{R}$ is discontinuous on $\mathbb Q$. Is $f$ necessarily discontinuous on $\mathbb R \setminus \mathbb Q$?

By Baire's theorem exotic examples are even the majority:

An explicit example is Thomae's function.

As another example take: $$f(x\in\mathbb{Q}):=(x-\pi)^2\quad f(x\
otin\mathbb{Q}):=-(x-\pi)^2$$ _(That one is even differentiable only at Pi but discontinuous everywhere else.)_