Does there existsany diffentiable function $f$ such that $f'$ is discontinuous exactly on $\Bbb{Q} $ and continuous on $\Bbb{R}\setminus \Bbb{Q}$ ?
Since $\Bbb{Q}$ is $F_{\sigma}$ , we can produce a function which is discontinuous only on $\Bbb{Q}$.
For an example we can pick Thomae's function.But Thomae's function has no primitive. Because if thomae's function $f$ has a primitive $F$ then $F'=f $ . Since $F'$ is Darboux function, image of $F'=f$ must contains an intervals and this is not possible as Thomae's function doesn't attain irrational values.
By choosing a particular example, we can conclude the impossibility of existence of such function.
If $f'$ is Darboux function and belongs to Baire class $1$ then $f'$ has a primitive $f$ .
Hence our goal is to create a Darboux function $f'$ of Baire class $1$ which is continuous on $\Bbb{Q}$ and discontinuous on $\Bbb{R}\setminus \Bbb{Q}$.
How to produce such function?