Riemann integrability; discontinuity

Question

I know that any Riemann integrable functions have at most countably many discontinuities, and similarly, any bounded functions with countable number of discontinuities are Lebesgue integrable. But is it necessarily true that any Riemann integrable functions have at most countably many discontinuities?

Thanks

Answer 1

No, it is not true. There is a theorem by Lebesgue that shows that a function is Riemman integrable if and only if its set of discontinuity points has Lebesgue measure zero. In particular, it can be uncountable. One such function is the characteristic function of the complement of the Cantor set.