Question: Is the set of points of continuity of any Riemann integrable function uncountable?
There's a question in my Analysis assignment asking us to prove if $f$ is integrable then it has infinitely many points of continuity (and the set is dense). I wonder if the set is also uncountable. I've seen examples of Riemann integrable functions that have uncountably many discontinuities but I haven't seen an example with only countably many points of continuity.
Best Answer
The set of points of discontinuity has Lebesgue measure $0$ (by to a theorem due to Lebesgue). So, the set of points of continuity has Lebesgue measure greater than $0$, and therefore it is uncountable.