Mixing
Choosing Partitions When Proving Integrability
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When I am trying to prove a function is integrable, is there a trick to choosing the partition that will work? I see that sometimes epsilon is used and sometimes delta is used but which version shall I choose in different scenarios?
(Aside: I usually use the fact that $U(f, P) - L(f, P) < epsilon$ as my proof)
calculus integration
asked Jan 28 at 5:32
Emma PascoeEmma Pascoe
161
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add a comment |
$begingroup$
When I am trying to prove a function is integrable, is there a trick to choosing the partition that will work? I see that sometimes epsilon is used and sometimes delta is used but which version shall I choose in different scenarios?
(Aside: I usually use the fact that $U(f, P) - L(f, P) < epsilon$ as my proof)
calculus integration
asked Jan 28 at 5:32
Emma PascoeEmma Pascoe
161
$endgroup$
add a comment |
$begingroup$
When I am trying to prove a function is integrable, is there a trick to choosing the partition that will work? I see that sometimes epsilon is used and sometimes delta is used but which version shall I choose in different scenarios?
(Aside: I usually use the fact that $U(f, P) - L(f, P) < epsilon$ as my proof)
calculus integration
asked Jan 28 at 5:32
Emma PascoeEmma Pascoe
161
$endgroup$
When I am trying to prove a function is integrable, is there a trick to choosing the partition that will work? I see that sometimes epsilon is used and sometimes delta is used but which version shall I choose in different scenarios?
(Aside: I usually use the fact that $U(f, P) - L(f, P) < epsilon$ as my proof)
calculus integration
calculus integration
asked Jan 28 at 5:32
Emma PascoeEmma Pascoe
161
asked Jan 28 at 5:32
Emma PascoeEmma Pascoe
161
asked Jan 28 at 5:32
Emma PascoeEmma Pascoe
161
asked Jan 28 at 5:32
Emma PascoeEmma Pascoe
161
asked Jan 28 at 5:32
Emma PascoeEmma Pascoe
161
161
add a comment |
add a comment |
1 Answer
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It's hard to say, it depends on the function; but the general idea is this: your condition $U(f, P) - L(f, P) < epsilon$ can be written as $sum_{i = 1}^n(M_i - m_i) Delta x_i< epsilon$. You want to make the LHS small, which you can do by either making $M_i - m_i$ small or $delta x_i$ small. So if there are intervals over which you know a strong thing about the function (such as it's uniformly continuous there) you want to create the partition so that it takes advantage of that. This would qualify as making $M_i - m_i$ small.
In the remaining intervals, where the function is more crazy, you can just make the $Delta x_i$ small. You can see an example of this in Rudin's proof that composition of a continuous function with an integrable function is integrable.
answered Jan 28 at 6:09
OviOvi
12.4k1040114
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1 Answer
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$begingroup$
It's hard to say, it depends on the function; but the general idea is this: your condition $U(f, P) - L(f, P) < epsilon$ can be written as $sum_{i = 1}^n(M_i - m_i) Delta x_i< epsilon$. You want to make the LHS small, which you can do by either making $M_i - m_i$ small or $delta x_i$ small. So if there are intervals over which you know a strong thing about the function (such as it's uniformly continuous there) you want to create the partition so that it takes advantage of that. This would qualify as making $M_i - m_i$ small.
In the remaining intervals, where the function is more crazy, you can just make the $Delta x_i$ small. You can see an example of this in Rudin's proof that composition of a continuous function with an integrable function is integrable.
answered Jan 28 at 6:09
OviOvi
12.4k1040114
$endgroup$
add a comment |
$begingroup$
It's hard to say, it depends on the function; but the general idea is this: your condition $U(f, P) - L(f, P) < epsilon$ can be written as $sum_{i = 1}^n(M_i - m_i) Delta x_i< epsilon$. You want to make the LHS small, which you can do by either making $M_i - m_i$ small or $delta x_i$ small. So if there are intervals over which you know a strong thing about the function (such as it's uniformly continuous there) you want to create the partition so that it takes advantage of that. This would qualify as making $M_i - m_i$ small.
In the remaining intervals, where the function is more crazy, you can just make the $Delta x_i$ small. You can see an example of this in Rudin's proof that composition of a continuous function with an integrable function is integrable.
answered Jan 28 at 6:09
OviOvi
12.4k1040114
$endgroup$
add a comment |
$begingroup$
It's hard to say, it depends on the function; but the general idea is this: your condition $U(f, P) - L(f, P) < epsilon$ can be written as $sum_{i = 1}^n(M_i - m_i) Delta x_i< epsilon$. You want to make the LHS small, which you can do by either making $M_i - m_i$ small or $delta x_i$ small. So if there are intervals over which you know a strong thing about the function (such as it's uniformly continuous there) you want to create the partition so that it takes advantage of that. This would qualify as making $M_i - m_i$ small.
In the remaining intervals, where the function is more crazy, you can just make the $Delta x_i$ small. You can see an example of this in Rudin's proof that composition of a continuous function with an integrable function is integrable.
answered Jan 28 at 6:09
OviOvi
12.4k1040114
$endgroup$
It's hard to say, it depends on the function; but the general idea is this: your condition $U(f, P) - L(f, P) < epsilon$ can be written as $sum_{i = 1}^n(M_i - m_i) Delta x_i< epsilon$. You want to make the LHS small, which you can do by either making $M_i - m_i$ small or $delta x_i$ small. So if there are intervals over which you know a strong thing about the function (such as it's uniformly continuous there) you want to create the partition so that it takes advantage of that. This would qualify as making $M_i - m_i$ small.
In the remaining intervals, where the function is more crazy, you can just make the $Delta x_i$ small. You can see an example of this in Rudin's proof that composition of a continuous function with an integrable function is integrable.
answered Jan 28 at 6:09
OviOvi
12.4k1040114
answered Jan 28 at 6:09
OviOvi
12.4k1040114
answered Jan 28 at 6:09
OviOvi
12.4k1040114
answered Jan 28 at 6:09
OviOvi
12.4k1040114
12.4k1040114
add a comment |
add a comment |
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