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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?

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

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    I don't see where choice comes in for the classic $h(x)=0$ on the rationals, $1$ on the irrationals, but I'm not an expert.2012-09-12