I understand that it follows from the first fundamental theorem of calculus that if a function is continuous, it has a primitive function and the Newton's and Riemann's integral have the same value.
What about non-continuous functions? Is it true that if both integrals exist, they are the same? It can obviously happen that function is Newton integrable but not Riemann integrable (e.g. unbounded functions). What about the other way? Can a function be Riemann integrable but not Newton integrable?
I looked up many questions on this site but found nothing.