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October 19th, 2016, 09:08 AM  #1 
Newbie Joined: Oct 2016 From: Netherlands Posts: 3 Thanks: 0  product of two functions as a subset of the union
Hi all, Setting: f,g A > R integrable My question is: why is it true that the set {x in A  (f*g)(x) is not continuous} is a subset of {x in A  f(x) is not continuous} united with {x in A  g(x) is not continuous} ? I'm trying to solve problem 3.14 of Spivak's Calculus on Manifolds. I have to show that when f and g are integrable so is the product of them. Greetings and thanks, T 
October 19th, 2016, 11:44 AM  #2 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,158 Thanks: 90 
If f and g are integrable they have a finite number of discontinuities and isolated points at infinity. Then fg has a finite number of discontinuities and isolated points at infinity and so is also integrable. 
October 19th, 2016, 02:33 PM  #3 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,158 Thanks: 90 
f is integrable if $\displaystyle \sum f_{i} \Delta x_{i}$ exists and is unique for arbitrary subdivisions approaching zero. $\displaystyle g_{i} \Delta x_{i}$ is an arbitrary subdivision approaching zero. Therefore, $\displaystyle \sum f_{i}(g_{i} \Delta x_{i})$ exists and is unique. 
November 2nd, 2016, 10:27 PM  #4  
Senior Member Joined: Sep 2016 From: USA Posts: 169 Thanks: 79 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
As for the original question, does f*g mean the pointwise product or the convolution? Assuming the former, I suggest you try proving this first in the case that both $f$ and $g$ are monotone functions. This seems much easier and once done you can use the result that integrable functions can be uniformly approximated by compactly supported functions with bounded variation which can be decomposed as the difference of two monotone functions. It is worth noting (and may be helpful) that monotone functions have at most countably many discontinuities and they are all jump type.  

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continuous, functions, product, subset, union 
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