Discontinuous Functions on the Real Line

Question

I want to prove that given any function $g:\mathbb{Z} \to \mathbb{R}$ there exists a function $f:\mathbb{R}\to\mathbb{R}$ such that its restriction to the integers is equal to g and such that it not continuous in any restriction of the domain to any open set in $\mathbb{R}$. I want to prove that such a function exists in ZF Set theory. My math teacher said I would probably need the axiom of choice. Could someone help me?

Answer 1

Hint: do you know any functions $\mathbb R \to \mathbb R$that are discontinuous everywhere? Can you adapt one to this purpose?