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I have examples where the set of discontinuities is $\mathbb{Q}$, so is neither open or closed. But are there other examples? I haven't found more.

Lore Rodriguez
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  • I don't understand your question. You already know that it isn't open or closed. What sort of answer are you expecting? Have you looked at this article? – Erick Wong May 10 '16 at 16:39
  • I wanted to know if there are other examples not with an open and closed set of discontinuities at the same time – Lore Rodriguez May 10 '16 at 16:42
  • Thanks for clarifying, I suggest you edit the title of the equation to something like "Examples of functions where the set of discontinuities is not open or closed". – Erick Wong May 10 '16 at 16:43
  • If you wish to ask about "other examples", you should clarify what examples (of functions) you know about, and perhaps sharpen the Question to indicate whether other functions with set of discontinuities $\mathbb{Q}$ are of interest. – hardmath May 10 '16 at 19:23
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    The set of discontinuities of a function $f:\mathbb R\to\mathbb R$ is always an $F_\sigma$ set (countable union of closed sets), and every $F_\sigma$ subset of $\mathbb R$ is the set of discontinuities of some function $f:\mathbb R\to\mathbb R.$ – bof May 10 '16 at 19:38
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    For a precise characterization (which I notice @bof posted while I was hunting down a reference), see the mathoverflow question Find a continuous function with a prescribed continuity set. – Dave L. Renfro May 10 '16 at 19:41

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