(Riemann) Integrability under composition of functionsby Mathmos6 Tags: composition, functions, integrability, riemann 

#1
Mar2509, 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 nonintegrable? 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 



#2
Mar2509, 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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