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Let $f:\mathbb{R}\to\mathbb{R}$. Let $S$ be the set of points at which $f$ is discontinuous. Which of the following could be $S$?

  1. All positive real numbers
  2. All irrational numbers
  3. All real numbers

I know that 3 is possible. One example is Dirichlet function. But I am not sure about 1 and 2. For 2, I know that there is no function $g$ on $[0,1]$ such that $g$ is continuous only at rational points. So I suspect 2 to be impossible. For 3, I am not sure but I suspect that something could go wrong near zero. Namely, if $f$ is continuous at zero, $f$ is continuous near zero.

user34183
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  • The answer for (2) is no. http://math.stackexchange.com/questions/67620/set-of-continuity-points-of-a-real-function – Simon Oct 22 '16 at 18:09
  • The set of points of continuity of a function must be a $G_\delta$ set, but the rational numbers is not a $G_\delta$ set. So there is no real function continuous at only the rational numbers, and thus none who are discontinuous at only the irrational numbers. – pancini Oct 22 '16 at 18:11

1 Answers1

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Define $$f(x)=\begin{cases} x,&\text{ if }x\text{ is positive and irrational}, \\0,&\text{ otherwise}.\end{cases}$$

Then $f$ is discontinuous at precisely the positive numbers.

Mitchell Spector
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