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Prove or disprove: if the set of continuous points of function $f\colon \mathbb{R}\to\mathbb{R}$ is dense everywhere, then

  1. the set of continuous points of $f$ is uncountable;

  2. the set of discontinuous points of $f$ is countable.

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    @today: What have you tried?2011-05-24
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    today-: Have you seen that the set of points of discontinuity of a function must be an $F_\sigma$? This could help you answer 1. The converse could help with 2, but it can be answered more directly. Do you have any thoughts to add, such as where you got stuck?2011-05-24
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    @today: Show that the set of points of continuity is $G_{\delta}$2011-05-24
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    @today: Please see: http://en.wikipedia.org/wiki/G%CE%B4_set2011-05-24
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    today-: Welcome to the website. Around here, people are usually happier to invest time to help if you give more than just a problem statement. You could ask particular *questions*, and indicating difficulties you've had (or any thoughts at all) would make it easier for people to direct their answers to be more helpful. You can edit your question by clicking "edit" in the lower left.2011-05-24
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    Wiki is good enough, thank guys above!2011-05-24
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    @today: That's good to hear! If you're willing to elaborate here on how your question has been answered, it could still be of benefit. For one thing, users here could help you clear up any possible remaining doubts. For another, it would benefit future readers interested in the question.2011-05-24
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    Obviously, if $2$ is true, then $1$ is true.2011-05-24
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    @Thomas: True, but of course 2 isn't :).2011-05-24

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