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 \(|l|<x, \space |f(n+l)-f(n)|<\epsilon\)