Is there a function that is continuous at every irrational but discontinuous at rational? Does there exist a function $f$ such that it is discontinuous at rational and continuous at every irrational but the restriction on $f$ to the set of all irrational numbers is not constant and $f(q_n)$ is convergent where $\{q_n\}$ is a sequence of rationals.
Thomae function is one of an example of function that satisfies above condition if the restriction $f$ to the set of all irrational numbers is constant.

Denote by $T(x)$ Thomae's function. Then
$$f(x)=T(x)+x$$
is a function satisfying your condition.
