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What is an example of a function which is not Riemann integrable on [0,1] but for which |f| is Riemann integrable on [0,1]? What is the upper integral and the lower integral of this function f, which are not equal, thus not Riemann integrable?

Remy
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1 Answers1

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Take the function $f$ equal to $1$ for $x$ rational and to $-1$ for $ x$ irrational. Upper integral is $1$ and lower integral $-1$.

mathcounterexamples.net
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  • Thanks, that makes sense, as |f| would just be y=1. I am a little confused about how to calculate the upper and lower integral. For this example, it seemed intuitive, but others, it is more complex, where you have to take partitions and sup's and make some calculations. – Remy Nov 21 '16 at 21:04
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    The upper integral is equal to $1$ as all open interval contains a rational number (the rationals are dense in $\mathbb R$). Hence the supremum of $f$ on all open interval is equal to $1$. – mathcounterexamples.net Nov 21 '16 at 21:07