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Suppose $C \subset \mathbb{R}$ is of type $F_{\sigma}$. That is $C$ can be written as the union of $F_{n}$'s where each $F_{n}$'s are closed. Then can we prove that each point of $C$ is a point of discontinuity for some $f: \mathbb{R} \to \mathbb{R}$.

I refered this link on wiki : http://en.wikipedia.org/wiki/Thomae%27s_function and in the follow up subsection they given this result. I would like somebody to explain it more precisely.

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Please fix the link. –  AD. Aug 28 '10 at 4:20
    
You need to say something more if you want an interesting problem. Maybe you want the $F_n$'s to be small in some sense? –  AD. Aug 28 '10 at 4:29
    
@Kenny TM: Thanks for fixing the link! –  anonymous Aug 28 '10 at 7:59
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up vote 2 down vote accepted

I believe you're looking for something like the construction mentioned here.

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One can also see this article, "S.S.Kim, Amer.Math.Monthly 106 (1999), 258-259".

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