Does any example of this particular type of function, stated below exist? f:$R\to R$ is continuous at every rational numbers and discontinuous at every irrational points. I can't find any example. If there does not exist any such function then why?
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The set of continuity points is a $G_{\delta}$ set, countable intersection of open sets, rationals aren't. – lulu Sep 23 '15 at 13:05
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As a reference to that fact: http://math.stackexchange.com/questions/138072/how-to-show-that-the-set-of-points-of-continuity-is-a-g-delta – lulu Sep 23 '15 at 13:07
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is the set of irrational numbers $G_\delta$ set? – Rupsa Sep 23 '15 at 13:08
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Yes. They are the intersection of ${\mathbb R - q_i}$ where the $q_i's$ list the rationals. – lulu Sep 23 '15 at 13:09
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By the way, that proves that $\mathbb Q$ is not $G_{\delta}$, for it were the countable intersection of open sets $O_i$ then we could intersect these with the open sets ${\mathbb R - q_i}$ to get $\emptyset$ as a countable intersection of dense open sets, in violation of the Baire category theorem. – lulu Sep 23 '15 at 13:12
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The link suggested by @MiguelAtencia covers all this very clearly. – lulu Sep 23 '15 at 13:14