Example for integrable function $f(x,y)$ such that $f_x$ is not integrable
0
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Is there an example for integrable function $f(x,y)$ such that $f_x$ is not integrable.
Here $f_x(y)=f(x,y)$ Is a function of only one variable $y$, means that $x$ Is a constant.
I know that from Fubini's theorem, $f_x$ cannot be integrable only on a set of measure $0$.
So is such an example exists?
real-analysis integration multivariable-calculus
asked Dec 26 '18 at 9:59
Gabi GGabi G
442110
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add a comment |
0
$begingroup$
Is there an example for integrable function $f(x,y)$ such that $f_x$ is not integrable.
Here $f_x(y)=f(x,y)$ Is a function of only one variable $y$, means that $x$ Is a constant.
I know that from Fubini's theorem, $f_x$ cannot be integrable only on a set of measure $0$.
So is such an example exists?
real-analysis integration multivariable-calculus
asked Dec 26 '18 at 9:59
Gabi GGabi G
442110
$endgroup$
1
$begingroup$
$f(x,y)=1$ if $xne 0$ and $f(0,y)=D(y)$ - Dirichlet-function
$endgroup$
– Minz
Dec 26 '18 at 10:08
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Are you sure $f(x,y)$ Is integrable?
$endgroup$
– Gabi G
Dec 26 '18 at 10:15
$begingroup$
It because the set of non continuity points are of measure $0$, right?
$endgroup$
– Gabi G
Dec 26 '18 at 10:17
1
$begingroup$
@Gabi: As an easier example, you can just take $f = 1_{{0} times Bbb{R}}$.
$endgroup$
– PhoemueX
Dec 26 '18 at 10:19
$begingroup$
Thanks to you both!
$endgroup$
– Gabi G
Dec 26 '18 at 10:25
add a comment |
0
0
0
$begingroup$
Is there an example for integrable function $f(x,y)$ such that $f_x$ is not integrable.
Here $f_x(y)=f(x,y)$ Is a function of only one variable $y$, means that $x$ Is a constant.
I know that from Fubini's theorem, $f_x$ cannot be integrable only on a set of measure $0$.
So is such an example exists?
real-analysis integration multivariable-calculus
asked Dec 26 '18 at 9:59
Gabi GGabi G
442110
$endgroup$
Is there an example for integrable function $f(x,y)$ such that $f_x$ is not integrable.
Here $f_x(y)=f(x,y)$ Is a function of only one variable $y$, means that $x$ Is a constant.
I know that from Fubini's theorem, $f_x$ cannot be integrable only on a set of measure $0$.
So is such an example exists?
real-analysis integration multivariable-calculus
real-analysis integration multivariable-calculus
asked Dec 26 '18 at 9:59
Gabi GGabi G
442110
asked Dec 26 '18 at 9:59
Gabi GGabi G
442110
asked Dec 26 '18 at 9:59
Gabi GGabi G
442110
asked Dec 26 '18 at 9:59
Gabi GGabi G
442110
asked Dec 26 '18 at 9:59
Gabi GGabi G
442110
442110
1
$begingroup$
$f(x,y)=1$ if $xne 0$ and $f(0,y)=D(y)$ - Dirichlet-function
$endgroup$
– Minz
Dec 26 '18 at 10:08
$begingroup$
Are you sure $f(x,y)$ Is integrable?
$endgroup$
– Gabi G
Dec 26 '18 at 10:15
$begingroup$
It because the set of non continuity points are of measure $0$, right?
$endgroup$
– Gabi G
Dec 26 '18 at 10:17
1
$begingroup$
@Gabi: As an easier example, you can just take $f = 1_{{0} times Bbb{R}}$.
$endgroup$
– PhoemueX
Dec 26 '18 at 10:19
$begingroup$
Thanks to you both!
$endgroup$
– Gabi G
Dec 26 '18 at 10:25
add a comment |
1
$begingroup$
$f(x,y)=1$ if $xne 0$ and $f(0,y)=D(y)$ - Dirichlet-function
$endgroup$
– Minz
Dec 26 '18 at 10:08
$begingroup$
Are you sure $f(x,y)$ Is integrable?
$endgroup$
– Gabi G
Dec 26 '18 at 10:15
$begingroup$
It because the set of non continuity points are of measure $0$, right?
$endgroup$
– Gabi G
Dec 26 '18 at 10:17
1
$begingroup$
@Gabi: As an easier example, you can just take $f = 1_{{0} times Bbb{R}}$.
$endgroup$
– PhoemueX
Dec 26 '18 at 10:19
$begingroup$
Thanks to you both!
$endgroup$
– Gabi G
Dec 26 '18 at 10:25
1
1
$begingroup$
$f(x,y)=1$ if $xne 0$ and $f(0,y)=D(y)$ - Dirichlet-function
$endgroup$
– Minz
Dec 26 '18 at 10:08
$begingroup$
$f(x,y)=1$ if $xne 0$ and $f(0,y)=D(y)$ - Dirichlet-function
$endgroup$
– Minz
Dec 26 '18 at 10:08
$begingroup$
Are you sure $f(x,y)$ Is integrable?
$endgroup$
– Gabi G
Dec 26 '18 at 10:15
$begingroup$
Are you sure $f(x,y)$ Is integrable?
$endgroup$
– Gabi G
Dec 26 '18 at 10:15
$begingroup$
It because the set of non continuity points are of measure $0$, right?
$endgroup$
– Gabi G
Dec 26 '18 at 10:17
$begingroup$
It because the set of non continuity points are of measure $0$, right?
$endgroup$
– Gabi G
Dec 26 '18 at 10:17
1
1
$begingroup$
@Gabi: As an easier example, you can just take $f = 1_{{0} times Bbb{R}}$.
$endgroup$
– PhoemueX
Dec 26 '18 at 10:19
$begingroup$
@Gabi: As an easier example, you can just take $f = 1_{{0} times Bbb{R}}$.
$endgroup$
– PhoemueX
Dec 26 '18 at 10:19
$begingroup$
Thanks to you both!
$endgroup$
– Gabi G
Dec 26 '18 at 10:25
$begingroup$
Thanks to you both!
$endgroup$
– Gabi G
Dec 26 '18 at 10:25
add a comment |
0
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1
$begingroup$
$f(x,y)=1$ if $xne 0$ and $f(0,y)=D(y)$ - Dirichlet-function
$endgroup$
– Minz
Dec 26 '18 at 10:08
$begingroup$
Are you sure $f(x,y)$ Is integrable?
$endgroup$
– Gabi G
Dec 26 '18 at 10:15
$begingroup$
It because the set of non continuity points are of measure $0$, right?
$endgroup$
– Gabi G
Dec 26 '18 at 10:17
1
$begingroup$
@Gabi: As an easier example, you can just take $f = 1_{{0} times Bbb{R}}$.
$endgroup$
– PhoemueX
Dec 26 '18 at 10:19
$begingroup$
Thanks to you both!
$endgroup$
– Gabi G
Dec 26 '18 at 10:25