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(Riemann) Integrability under composition of functions

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Mathmos6
#1
Mar25-09, 06:39 AM
P: 82
1. The problem statement, all variables and given/known data
I've been looking at how integrable functions behave under composition, and I know that if f and g are integrable, f(g(x)) is not necessarily integrable, but it -is- necessarily integrable if f is continuous, regardless of whether g is. So I was wondering, what about if g is continuous and f wasn't, rather than f was? Is there an example of a discontinuous integrable f and a continuous integrable g such that f(g(x)) is non-integrable?

I don't want an explicit proof of the answer, but I'd just like to know whether there is or isn't such an example, so I can begin looking for a counterexample or a proof as appropriate, rather than ending up trying to prove something which is false or look for a counterexample for something which is true. My intuition tells me there isn't a counterexample, but I've found relying on intuition in analysis is a very bad idea!

Thanks,

Mathmos6
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Wretchosoft
#2
Mar25-09, 12:58 PM
P: 64
No, there is no counterexample, and the proof is relatively simple (aka don't try too hard.).


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