# Is the Lebesgue measure zero for the discontinuous set of a semicontinuous function?

[Q.] Is there a semicontinuous function, which has its discontinuous set with non-zero measure?

Remark: Given a semicontinuous function, the set of all discontinuous points may be uncountable, for instance, an indicator function on a Cantor set.

I also found a paper here with its title: "Baire order of functions are continuous almost everywhere", see this link:

http://www.ams.org/journals/proc/1975-051-02/S0002-9939-1975-0372128-1/S0002-9939-1975-0372128-1.pdf

I do not understand this paper, but does it say all semicontinuous functions are continuous almost everywhere, since it is baire-1 function?

## migrated from mathoverflow.netJun 21 '16 at 21:56

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• There are Cantor sets of positive measure. You simply take out less then one third in every step. – Zero Jun 21 '16 at 14:45
• @Anton Thanks for your answer. – user79963 Jun 21 '16 at 14:52
• The paper's title is not copied correctly and your statement for what the paper says is not correct. The result you want is: If $f:{\mathbb R} \rightarrow {\mathbb R}$ is semicontinuous, then the discontinuity set of $f$ is $F_{\sigma}$ and meager (i.e. first Baire category) in the reals. Conversely, any subset of the reals having these two properties is the discontinuity set for some semicontinuous function $f:{\mathbb R} \rightarrow {\mathbb R}.$ For references to this result, see Theorem 3' here. – Dave L. Renfro Jun 21 '16 at 19:09