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Show that if $f$ is Riemann integrable on $[a,b]$ then $|f|$ is also Riemann integrable on $[a,b]$.

My idea is: let $f$ be in $[a,b]$ less than $|f|$, since $f$ is integrable then $|f|$ is also integrable on $[a,b]$.

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    This is a direct consequence of the Lebesgue's integrability criterion. See here: http://en.wikipedia.org/wiki/Riemann_integral#Integrability.2012-01-06
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    But this problem is often stated to serve as an exercise in the use and understanding of Riemann sums.2012-01-06
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    I think my answer [here](http://math.stackexchange.com/q/84468/8271) can be useful to you.2012-01-06
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    The converse holds if $f$ is a derivative: http://math.stackexchange.com/questions/933702012-01-06

3 Answers 3