I have a following question in my mind regarding the set of continuity of points of $f$:
Does there exist a real valued function on $[0,1]$ such that the set of continuity points of $f$ are the set of algebraic numbers in $[0,1]$?
Thank You.
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3See this and this. – David Mitra Nov 07 '16 at 10:27
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Nope: this is because algebraic numbers are not a $G_{\delta}$ set. – Crostul Nov 07 '16 at 10:28