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I want a proof for this theorem:

Let $f$ be a function on $[a,b]$. Then $f$ is Riemann integrable if and only if $f$ is bounded and continuous almost everywhere.

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I think you mean Riemann. –  glebovg Nov 15 '12 at 18:34

2 Answers 2

This is known as the Lebesgue criterion for Riemann integrability. You can Google it or see a similar question and the wiki article for proof.

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See Munkres - Analysis on manifolds for example.

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an answer should consist of more than simply a link or reference (unless the question is a request for a reference). –  robjohn Nov 17 '12 at 5:38

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