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Assume you have a Riemann-Integrable function $f$ on $[a,b]$. Then we know that it is Lebesgue-measurable, and it is continuous Lebesgue almost everywhere.

Question 1: Will the set of discontinuities be Borel-measurable?

Question 2: If we in addition assume that $f$ is Borel-measurable, will then the set of discontinuities be Borel-measurable?

user394334
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    The set of discontinuities will be $F_{\sigma}$, hence Borel-measurable. This holds for any function $\mathbb{R}\rightarrow\mathbb{R}$ and has nothing to do with Riemann-integrability. – Thorgott Jan 31 '20 at 21:41
  • https://math.stackexchange.com/questions/2544162/discontinuity-set-of-function-closed-and-f-sigma – M A Pelto Jan 31 '20 at 22:10
  • @Thorgott You should add that as an answer. – Noah Schweber Jan 31 '20 at 23:58

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