Flashcards: Riemann Integral, Riemann Sums, & Improper Riemann Integration

What conditions are necessary to prove that the upper and lower integrals of a bounded function exist?

\(\displaystyle a\)\(\displaystyle b\ \epsilon\ \mathbb{R}\)\(\displaystyle a< b\),  and \(\displaystyle f:[a,b]\rightarrow \mathbb{R}\) be bounded

\(\displaystyle a\)\(\displaystyle b\ \epsilon\ \mathbb{R}\)\(\displaystyle a< b\),  and \(\displaystyle f:[a,b]\rightarrow \mathbb{Z}\) be bounded

\(\displaystyle a\)\(\displaystyle b\ \epsilon\ \mathbb{R}\)\(\displaystyle a>b\),  and \(\displaystyle f:[a,b]\rightarrow \mathbb{R}\)

\(\displaystyle a\)\(\displaystyle b\ \epsilon\ \mathbb{Z}\)\(\displaystyle a< b\),  and \(\displaystyle f:[a,b]\rightarrow \mathbb{R}\) be bounded

\(\displaystyle a\)\(\displaystyle b\ \epsilon\ \mathbb{R}\)\(\displaystyle a>b\),  and \(\displaystyle f:[a,b]\rightarrow \mathbb{Z}\)

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