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This is an exercise from A second course on real functions.

Do you have an example of a semicontinuous function defined on $[0,1]$ which is discontinuous at an uncountable number of points?

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    $\begingroup$ A characteristic function of a famous set. $\endgroup$ – Daniel Fischer Sep 20 '15 at 17:39
  • $\begingroup$ It is even possible to make a stronger version of the statement that the set of discontinuity has a positive Lebesgue measure. Just take the indicator of a Cantor set on $[0,1]$ of positive measure. $\endgroup$ – A.Γ. Sep 20 '15 at 18:41
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Indicator function on Cantor set, as suggested by @Daniel Fischer's comment.

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