So we know a function has set of points of discontinuity $F_{\sigma}$.
Question: Is any $F_{\sigma}$ the set of discontinuity of some $f$ ?
Tried but failed. $\text{Thank you very much}$.
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You actually want to know : does it true that for some $f$ there is a set of points of dicontinuity? – openspace Jan 12 '17 at 20:22
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@openspace I want to know if given two metric spaces $(X,d)$ and $(Y,d')$ and $F$ a $F_{\sigma}$ subset of $X$, there is a function $f:X\to Y$ for which ${x\in X:f\text{ is discontinuous at }x}=F$. If not in general then what about the case $X=Y=\mathbb{R}$ with usual metric? – NeedForHelp Jan 12 '17 at 20:25
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In general this is false (just think of a space with discrete metric). True for real numbers though. – Moishe Kohan Jan 12 '17 at 21:44
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1See my 20 December 2006 sci.math post References for Continuity Sets, especially the comments following the references listed under "PROOFS OF 2':". – Dave L. Renfro Jan 12 '17 at 21:45