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 kNiEsSoKk October 19th, 2016 10:08 AM

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

 zylo October 19th, 2016 12:44 PM

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.

 zylo October 19th, 2016 03:33 PM

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.

 SDK November 2nd, 2016 11:27 PM

Quote:
 Originally Posted by zylo (Post 551027) 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.
This is definitely not true.

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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