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Can an unbounded function in $R$ both continuous a dense set of points and discontinuous on a dense set of points?

I guess it is impossible since at least one discontinuity means the function is not continuous so I am looking for an example but can not find it.

TFC
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  • Do you know of a function that is continuous at the irrational points, and discontinuous at the rational points? – Calvin Lin Oct 27 '20 at 05:50
  • Hint: If you have a function $f$ such that $0<f<1$, $f$ is continuous on a dense set and discontinuous on another dense such that $f$ takes values arbitrarily close to $0$ then $\frac 1 f$ would be an example. – Kavi Rama Murthy Oct 27 '20 at 05:50
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    There are many such functions. One very common example is here. My answer to this question generalizes that example. There is a very different example here. – Brian M. Scott Oct 27 '20 at 05:54

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