What Would It Look Like?

Calculus Level 3

Does there exist a function that is continuous at all irrational points but not continuous at all rational points?

Definition (for this question): A function $f$ is said to be continuous at a point $n$ if for any positive $\epsilon$ there exists an $x$ such that for all real $|l|<x, \space |f(n+l)-f(n)|<\epsilon$