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. $\text{Thank you very much}$.
Discontinuous function on given set
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metric-spaces
discontinuous-functions
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0You actually want to know : does it true that for some $f$ there is a set of points of dicontinuity? – 2017-01-12
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0@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? – 2017-01-12
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0In general this is false (just think of a space with discrete metric). True for real numbers though. – 2017-01-12
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1See my [20 December 2006 sci.math post *References for Continuity Sets*](http://mathforum.org/kb/message.jspa?messageID=5447961), especially the comments following the references listed under "PROOFS OF 2':". – 2017-01-12