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Given $$ f(x) = \begin{cases} x, & \text{when x is rational} \\ -x, & \text{when x is not rational} \end{cases} $$ Show that $f$ is not Riemann-integrable over $[a,b]$, but $|f|$ is.

How to go about this problem? I started by taking a partition $P$ with each interval of equal length say $k$. Now clearly $U(P,f)$ is not equal to $L(P,f)$ and hence the limits. Is this approach fair?

mathematics2x2life
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user244722
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1 Answers1

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HINT. $\mathbb{Q}$ is dense in $\mathbb{R}$. So given an interval in the partition, what is its sup and inf over that interval? So what happens when you shrink down the intervals, taking the sum over all the intervals in your partition? Do the largest and smallest possible Riemann sums converge to each other?

For the second part, rewrite $f(x)$ using the definition of $f$ and the definition of $|\cdot|$. It should be clear what function you get and why it is integrable.

mathematics2x2life
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  • that's what I want someone to show. Clearly they won't I can feel but can't express. Should I take the length of each sub-interval to be same for easiness. – user244722 Feb 08 '20 at 17:26
  • @user244722 You will want to be more careful as if the integral exists, then both upper/lower limits will converge and then the integral will be independent of the partition (so long as it shrinks to 0 mesh). Hence why we typically choose the 'nice' evenly spaced one. But you won't be showing convergence. You can either leave the partition arbitrary and still arrive at the contradiction, or choose one carefully and show that sup/inf integrals do not approach each other under this partition, contradicting the definition of the R.I. – mathematics2x2life Feb 08 '20 at 17:35
  • sorry but I couldn't understand what you intend to say very clearly.. If you could show your work I would be grateful. – user244722 Feb 08 '20 at 17:40
  • @user244722 Perhaps it would be easier to see how you handle this in other similar cases, then adjusting the mesh argument to work for your problem, see then https://math.la.asu.edu/~kuiper/371files/ThomaeFunction.pdf and https://math.stackexchange.com/questions/1427612/integrability-of-thomaes-function-on-0-1 and http://homepages.rpi.edu/~olsond2/docs/Strange_Functions.pdf – mathematics2x2life Feb 08 '20 at 17:45
  • Downvoted because hints should go in comments. An answer should be a full solution. – gtoques Mar 18 '20 at 09:38