Example of a non-integrable function f , such that |f| and f^2 are integrable?

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A function f that meets the criteria of being discontinuous while having both |f| and f² integrable is f(x) = 1 for rational x and -1 for irrational x. This function is discontinuous everywhere, which aligns with the requirement for f. The discussion emphasizes that for Riemann or Newton integrals, such a function would not be integrable, but it is integrable in the broader sense. The participants acknowledge the challenge of finding such a function and express gratitude for the insights shared. Overall, the conversation highlights the nuances of integrability in mathematical analysis.
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I'm looking for a function
f:[a,b] -> R such that |f| and f2 are integrable on [a,b]
any helps?
 
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To start with, f must be discontinuous (a continuous function on a closed interval is integrable). Can you think of a function f such that f is discontinuous, but |f| (and thus f2 = |f|2) is continuous?This is really an analysis question, so it really belongs in that forum. :)[/size]
 
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f(x)= 1 for x rational,
-1 for x irational
 
Don't just give him an answer. ;)

(That's the exact same function I was thinking of, though.)
 
I forgot to say: Only if you are dealing with Newton or rieman integrals, otherwise f(x) is integrable.
 
Oh I don't know why I didn't think of that!
Thank you so much to both of youuuu! :smile:
 

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