Question

4.The necessary and sufficient conditions for a function to be Riemann integrable is that for every partition P ; which of the following conditions must occur​

Answer 1

In order for a function to be Riemann integrable on an interval first it has to be bounded. There are things called improper integrals that extend Riemann integrals to unbounded functions, but, strictly speaking, they’re not Riemann integrable.

Second, they have to have a limited set of discontinuities, more specifically, the set of discontinuities has to have Lebesgue measure zero. That’s called Lebesgue’s integrability condition.

Those two conditions are necessary, and they’re also sufficient as Lebesgue proved.

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