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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.

Phira
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mshj
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  • I think you mean Riemann. – glebovg Nov 15 '12 at 18:34
  • I put a note here coz I've added it to my favorite. By definition, any Riemann integrable function is bounded. Improper integral are considered as limit of Riemann integrable functions. So it will be more appropriate to correct the problem as following: Let f be a function on [a,b]. Then f is Riemann integrable if and only if f is bounded everywhere and continuous almost everywhere. – Bear and bunny Aug 04 '15 at 04:08
  • See Munkres - Analysis on manifolds for example. – dmm Nov 15 '12 at 18:36

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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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