# A semicontinuous function discontinuous at an uncountable number of points?

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?

• A characteristic function of a famous set. – Daniel Fischer Sep 20 '15 at 17:39
• 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. – A.Γ. Sep 20 '15 at 18:41