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Possible Duplicate:
Set of continuity points of a real function

Consider an function $f:{\mathbb R}\to{\mathbb R}$ and the standard topology. Suppose that $S\subset{\mathbb R}$ is the set of all the discontinuities of $f$. Here are my questions:

  • For any $S\subset{\mathbb R}$, can we always find an $f:{\mathbb R}\to{\mathbb R}$, such that $S$ is the set of all the discontinuities of $f$?
  • If the answer to the question above is NO, then what kinds of $S\subset{\mathbb R}$ generally can be?

I found several such $S\subset{\mathbb R}$:

  • $S = \emptyset:\qquad f(x) = x$
  • $S = {\mathbb R}$: $$ f(x) = \left\{ \begin{array}{ll} 0, & x\in{\mathbb Q}~; \\ 1, & x\in{\mathbb R}\setminus{\mathbb Q}~. \end{array} \right. $$
  • etc.

But I still don't have any idea how to answer the questions.

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    $S$ is the set of discontinuity of a real function if and only if it is an $F_\sigma$ (countable union of closed sets). See the thread above. – t.b. Oct 23 '11 at 03:12
  • @t.b.: Ah, I didn't notice that question. Thanks for you answer and closing the duplicate question. :-) –  Oct 23 '11 at 15:54

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