$frac1{x ^ 2 + y^2}$ is uniformly continuous in your domain? [closed]
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They let me see if the function$$frac1{x ^ 2 + y^2}$$ is uniformly continuous in their domain but I have not been able to solve the problem, will anyone have any suggestions on how to solve the problem?
multivariable-calculus multivalued-functions
edited Dec 26 '18 at 20:07
mechanodroid
28.7k62548
asked Dec 26 '18 at 19:51
Luis D. Barreto GarcíaLuis D. Barreto García
6
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closed as off-topic by RRL, zhw., KReiser, Paul Frost, egreg Dec 26 '18 at 23:55
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, KReiser, Paul Frost, egreg
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
They let me see if the function$$frac1{x ^ 2 + y^2}$$ is uniformly continuous in their domain but I have not been able to solve the problem, will anyone have any suggestions on how to solve the problem?
multivariable-calculus multivalued-functions
edited Dec 26 '18 at 20:07
mechanodroid
28.7k62548
asked Dec 26 '18 at 19:51
Luis D. Barreto GarcíaLuis D. Barreto García
6
$endgroup$
closed as off-topic by RRL, zhw., KReiser, Paul Frost, egreg Dec 26 '18 at 23:55
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, KReiser, Paul Frost, egreg
If this question can be reworded to fit the rules in the help center, please edit the question.
1
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What is the domain? The maximal one?
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– Arthur
Dec 26 '18 at 19:55
add a comment |
$begingroup$
They let me see if the function$$frac1{x ^ 2 + y^2}$$ is uniformly continuous in their domain but I have not been able to solve the problem, will anyone have any suggestions on how to solve the problem?
multivariable-calculus multivalued-functions
edited Dec 26 '18 at 20:07
mechanodroid
28.7k62548
asked Dec 26 '18 at 19:51
Luis D. Barreto GarcíaLuis D. Barreto García
6
$endgroup$
They let me see if the function$$frac1{x ^ 2 + y^2}$$ is uniformly continuous in their domain but I have not been able to solve the problem, will anyone have any suggestions on how to solve the problem?
multivariable-calculus multivalued-functions
multivariable-calculus multivalued-functions
edited Dec 26 '18 at 20:07
mechanodroid
28.7k62548
asked Dec 26 '18 at 19:51
Luis D. Barreto GarcíaLuis D. Barreto García
6
edited Dec 26 '18 at 20:07
mechanodroid
28.7k62548
asked Dec 26 '18 at 19:51
Luis D. Barreto GarcíaLuis D. Barreto García
6
edited Dec 26 '18 at 20:07
mechanodroid
28.7k62548
edited Dec 26 '18 at 20:07
mechanodroid
28.7k62548
edited Dec 26 '18 at 20:07
mechanodroid
28.7k62548
28.7k62548
asked Dec 26 '18 at 19:51
Luis D. Barreto GarcíaLuis D. Barreto García
6
asked Dec 26 '18 at 19:51
Luis D. Barreto GarcíaLuis D. Barreto García
6
asked Dec 26 '18 at 19:51
Luis D. Barreto GarcíaLuis D. Barreto García
6
6
closed as off-topic by RRL, zhw., KReiser, Paul Frost, egreg Dec 26 '18 at 23:55
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, KReiser, Paul Frost, egreg
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by RRL, zhw., KReiser, Paul Frost, egreg Dec 26 '18 at 23:55
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, KReiser, Paul Frost, egreg
If this question can be reworded to fit the rules in the help center, please edit the question.
1
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What is the domain? The maximal one?
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– Arthur
Dec 26 '18 at 19:55
add a comment |
1
$begingroup$
What is the domain? The maximal one?
$endgroup$
– Arthur
Dec 26 '18 at 19:55
1
1
$begingroup$
What is the domain? The maximal one?
$endgroup$
– Arthur
Dec 26 '18 at 19:55
$begingroup$
What is the domain? The maximal one?
$endgroup$
– Arthur
Dec 26 '18 at 19:55
add a comment |
1 Answer
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Assuming the domain is $mathbb{R}^2setminus {(0,0)}$, the answer is no.
Let $varepsilon > 0$ and consider the points $(x, 0)$ and $(x+varepsilon, 0)$ for some $x > 0$. We have
$$left|frac1{x^2} - frac1{(x+varepsilon)^2}right| = frac{2xvarepsilon + varepsilon^2}{(x+varepsilon)^2x^2}xrightarrow{xto 0} infty$$
but
$$|(x+varepsilon, 0) - (x, 0)| = varepsilon$$
We conclude that $(x,y) mapsto frac1{x^2+y^2}$ is not uniformly continuous.
answered Dec 26 '18 at 20:06
mechanodroidmechanodroid
28.7k62548
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add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Assuming the domain is $mathbb{R}^2setminus {(0,0)}$, the answer is no.
Let $varepsilon > 0$ and consider the points $(x, 0)$ and $(x+varepsilon, 0)$ for some $x > 0$. We have
$$left|frac1{x^2} - frac1{(x+varepsilon)^2}right| = frac{2xvarepsilon + varepsilon^2}{(x+varepsilon)^2x^2}xrightarrow{xto 0} infty$$
but
$$|(x+varepsilon, 0) - (x, 0)| = varepsilon$$
We conclude that $(x,y) mapsto frac1{x^2+y^2}$ is not uniformly continuous.
answered Dec 26 '18 at 20:06
mechanodroidmechanodroid
28.7k62548
$endgroup$
add a comment |
$begingroup$
Assuming the domain is $mathbb{R}^2setminus {(0,0)}$, the answer is no.
Let $varepsilon > 0$ and consider the points $(x, 0)$ and $(x+varepsilon, 0)$ for some $x > 0$. We have
$$left|frac1{x^2} - frac1{(x+varepsilon)^2}right| = frac{2xvarepsilon + varepsilon^2}{(x+varepsilon)^2x^2}xrightarrow{xto 0} infty$$
but
$$|(x+varepsilon, 0) - (x, 0)| = varepsilon$$
We conclude that $(x,y) mapsto frac1{x^2+y^2}$ is not uniformly continuous.
answered Dec 26 '18 at 20:06
mechanodroidmechanodroid
28.7k62548
$endgroup$
add a comment |
$begingroup$
Assuming the domain is $mathbb{R}^2setminus {(0,0)}$, the answer is no.
Let $varepsilon > 0$ and consider the points $(x, 0)$ and $(x+varepsilon, 0)$ for some $x > 0$. We have
$$left|frac1{x^2} - frac1{(x+varepsilon)^2}right| = frac{2xvarepsilon + varepsilon^2}{(x+varepsilon)^2x^2}xrightarrow{xto 0} infty$$
but
$$|(x+varepsilon, 0) - (x, 0)| = varepsilon$$
We conclude that $(x,y) mapsto frac1{x^2+y^2}$ is not uniformly continuous.
answered Dec 26 '18 at 20:06
mechanodroidmechanodroid
28.7k62548
$endgroup$
Assuming the domain is $mathbb{R}^2setminus {(0,0)}$, the answer is no.
Let $varepsilon > 0$ and consider the points $(x, 0)$ and $(x+varepsilon, 0)$ for some $x > 0$. We have
$$left|frac1{x^2} - frac1{(x+varepsilon)^2}right| = frac{2xvarepsilon + varepsilon^2}{(x+varepsilon)^2x^2}xrightarrow{xto 0} infty$$
but
$$|(x+varepsilon, 0) - (x, 0)| = varepsilon$$
We conclude that $(x,y) mapsto frac1{x^2+y^2}$ is not uniformly continuous.
answered Dec 26 '18 at 20:06
mechanodroidmechanodroid
28.7k62548
answered Dec 26 '18 at 20:06
mechanodroidmechanodroid
28.7k62548
answered Dec 26 '18 at 20:06
mechanodroidmechanodroid
28.7k62548
answered Dec 26 '18 at 20:06
mechanodroidmechanodroid
28.7k62548
28.7k62548
add a comment |
add a comment |
1
$begingroup$
What is the domain? The maximal one?
$endgroup$
– Arthur
Dec 26 '18 at 19:55