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Prove that the function $f(x,y)=frac{1}{sqrt[r]{x}(y^2-x^3)},r>frac{1}{2},y^2>x^3$ is bounded above on...
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How to prove function $f(x,y)=frac{1}{sqrt[r]{x}(y^2-x^3)},r>frac{1}{2},y^2>x^3$ is bounded above by some constant $k(r)$ (that is, $k$ depends upon $r$) on the subset $mathbb{Z}timesmathbb{Z}$ of $R^2$.
I cannot find many examples in books where two variable functions are proved to be bounded, however proving unboundedness is easy.
EDIT: I want the boundedness of $f$ on $mathbb{Z}timesmathbb{Z}$.
real-analysis analysis multivariable-calculus
asked Jan 28 at 1:24
ershersh
438113
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add a comment |
$begingroup$
How to prove function $f(x,y)=frac{1}{sqrt[r]{x}(y^2-x^3)},r>frac{1}{2},y^2>x^3$ is bounded above by some constant $k(r)$ (that is, $k$ depends upon $r$) on the subset $mathbb{Z}timesmathbb{Z}$ of $R^2$.
I cannot find many examples in books where two variable functions are proved to be bounded, however proving unboundedness is easy.
EDIT: I want the boundedness of $f$ on $mathbb{Z}timesmathbb{Z}$.
real-analysis analysis multivariable-calculus
asked Jan 28 at 1:24
ershersh
438113
$endgroup$
add a comment |
$begingroup$
How to prove function $f(x,y)=frac{1}{sqrt[r]{x}(y^2-x^3)},r>frac{1}{2},y^2>x^3$ is bounded above by some constant $k(r)$ (that is, $k$ depends upon $r$) on the subset $mathbb{Z}timesmathbb{Z}$ of $R^2$.
I cannot find many examples in books where two variable functions are proved to be bounded, however proving unboundedness is easy.
EDIT: I want the boundedness of $f$ on $mathbb{Z}timesmathbb{Z}$.
real-analysis analysis multivariable-calculus
asked Jan 28 at 1:24
ershersh
438113
$endgroup$
How to prove function $f(x,y)=frac{1}{sqrt[r]{x}(y^2-x^3)},r>frac{1}{2},y^2>x^3$ is bounded above by some constant $k(r)$ (that is, $k$ depends upon $r$) on the subset $mathbb{Z}timesmathbb{Z}$ of $R^2$.
I cannot find many examples in books where two variable functions are proved to be bounded, however proving unboundedness is easy.
EDIT: I want the boundedness of $f$ on $mathbb{Z}timesmathbb{Z}$.
real-analysis analysis multivariable-calculus
real-analysis analysis multivariable-calculus
asked Jan 28 at 1:24
ershersh
438113
asked Jan 28 at 1:24
ershersh
438113
edited Jan 28 at 1:46
ersh
asked Jan 28 at 1:24
ershersh
438113
asked Jan 28 at 1:24
ershersh
438113
asked Jan 28 at 1:24
ershersh
438113
438113
add a comment |
add a comment |
1 Answer
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The function is not bounded. Set y=1. Then if we look as x approaches 1 from below the function diverges to infinity.
answered Jan 28 at 1:32
Jagol95Jagol95
2637
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$begingroup$
Sorry, I want the boundedness of $f$ on $mathbb{Z}timesmathbb{Z}$. My mistake! I have edited my question.
$endgroup$
– ersh
Jan 28 at 1:47
$begingroup$
well in this case y^2-x^3 will be at least 1 and x^r certainly also can't get close to 0.
$endgroup$
– Jagol95
Jan 28 at 1:52
$begingroup$
So what would be the constant $k$?
$endgroup$
– ersh
Jan 28 at 1:57
add a comment |
Your Answer
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The function is not bounded. Set y=1. Then if we look as x approaches 1 from below the function diverges to infinity.
answered Jan 28 at 1:32
Jagol95Jagol95
2637
$endgroup$
$begingroup$
Sorry, I want the boundedness of $f$ on $mathbb{Z}timesmathbb{Z}$. My mistake! I have edited my question.
$endgroup$
– ersh
Jan 28 at 1:47
$begingroup$
well in this case y^2-x^3 will be at least 1 and x^r certainly also can't get close to 0.
$endgroup$
– Jagol95
Jan 28 at 1:52
$begingroup$
So what would be the constant $k$?
$endgroup$
– ersh
Jan 28 at 1:57
add a comment |
$begingroup$
The function is not bounded. Set y=1. Then if we look as x approaches 1 from below the function diverges to infinity.
answered Jan 28 at 1:32
Jagol95Jagol95
2637
$endgroup$
$begingroup$
Sorry, I want the boundedness of $f$ on $mathbb{Z}timesmathbb{Z}$. My mistake! I have edited my question.
$endgroup$
– ersh
Jan 28 at 1:47
$begingroup$
well in this case y^2-x^3 will be at least 1 and x^r certainly also can't get close to 0.
$endgroup$
– Jagol95
Jan 28 at 1:52
$begingroup$
So what would be the constant $k$?
$endgroup$
– ersh
Jan 28 at 1:57
add a comment |
$begingroup$
The function is not bounded. Set y=1. Then if we look as x approaches 1 from below the function diverges to infinity.
answered Jan 28 at 1:32
Jagol95Jagol95
2637
$endgroup$
The function is not bounded. Set y=1. Then if we look as x approaches 1 from below the function diverges to infinity.
answered Jan 28 at 1:32
Jagol95Jagol95
2637
answered Jan 28 at 1:32
Jagol95Jagol95
2637
answered Jan 28 at 1:32
Jagol95Jagol95
2637
answered Jan 28 at 1:32
Jagol95Jagol95
2637
2637
$begingroup$
Sorry, I want the boundedness of $f$ on $mathbb{Z}timesmathbb{Z}$. My mistake! I have edited my question.
$endgroup$
– ersh
Jan 28 at 1:47
$begingroup$
well in this case y^2-x^3 will be at least 1 and x^r certainly also can't get close to 0.
$endgroup$
– Jagol95
Jan 28 at 1:52
$begingroup$
So what would be the constant $k$?
$endgroup$
– ersh
Jan 28 at 1:57
add a comment |
$begingroup$
Sorry, I want the boundedness of $f$ on $mathbb{Z}timesmathbb{Z}$. My mistake! I have edited my question.
$endgroup$
– ersh
Jan 28 at 1:47
$begingroup$
well in this case y^2-x^3 will be at least 1 and x^r certainly also can't get close to 0.
$endgroup$
– Jagol95
Jan 28 at 1:52
$begingroup$
So what would be the constant $k$?
$endgroup$
– ersh
Jan 28 at 1:57
$begingroup$
Sorry, I want the boundedness of $f$ on $mathbb{Z}timesmathbb{Z}$. My mistake! I have edited my question.
$endgroup$
– ersh
Jan 28 at 1:47
$begingroup$
Sorry, I want the boundedness of $f$ on $mathbb{Z}timesmathbb{Z}$. My mistake! I have edited my question.
$endgroup$
– ersh
Jan 28 at 1:47
$begingroup$
well in this case y^2-x^3 will be at least 1 and x^r certainly also can't get close to 0.
$endgroup$
– Jagol95
Jan 28 at 1:52
$begingroup$
well in this case y^2-x^3 will be at least 1 and x^r certainly also can't get close to 0.
$endgroup$
– Jagol95
Jan 28 at 1:52
$begingroup$
So what would be the constant $k$?
$endgroup$
– ersh
Jan 28 at 1:57
$begingroup$
So what would be the constant $k$?
$endgroup$
– ersh
Jan 28 at 1:57
add a comment |
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