Does there exists a function which maps Cauchy sequence to Cauchy sequence but it isn't uniformly continuous...
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This question already has an answer here:
$f$ be a continuous function maps Cauchy into Cauchy. Is $f$ uniformly continuous?
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Does there exists a function which maps Cauchy sequence to Cauchy sequence but it isn't uniformly continuous?
we know uniformorly continuous function maps cauchy sequence to cauchy sequence.But my book doesn't say anything about converse of i think its false. but didn't find a counter example yet . please help!
real-analysis sequences-and-series cauchy-sequences
asked Jan 11 at 14:29
Cloud JRCloud JR
878517
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marked as duplicate by Martin R, Community♦ Jan 11 at 14:34
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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$begingroup$
This question already has an answer here:
$f$ be a continuous function maps Cauchy into Cauchy. Is $f$ uniformly continuous?
1 answer
Does there exists a function which maps Cauchy sequence to Cauchy sequence but it isn't uniformly continuous?
we know uniformorly continuous function maps cauchy sequence to cauchy sequence.But my book doesn't say anything about converse of i think its false. but didn't find a counter example yet . please help!
real-analysis sequences-and-series cauchy-sequences
asked Jan 11 at 14:29
Cloud JRCloud JR
878517
$endgroup$
marked as duplicate by Martin R, Community♦ Jan 11 at 14:34
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
2
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$f : x mapsto x^2$ on $Bbb R$
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– Chinnapparaj R
Jan 11 at 14:29
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Functions that map Cauchy sequences to Cauchy sequences are called Cauchy-continuous functitons. It's a nice little exercise to show that Cauchy-continuity is stronger than continuity.
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– Theo Bendit
Jan 11 at 14:36
add a comment |
$begingroup$
This question already has an answer here:
$f$ be a continuous function maps Cauchy into Cauchy. Is $f$ uniformly continuous?
1 answer
Does there exists a function which maps Cauchy sequence to Cauchy sequence but it isn't uniformly continuous?
we know uniformorly continuous function maps cauchy sequence to cauchy sequence.But my book doesn't say anything about converse of i think its false. but didn't find a counter example yet . please help!
real-analysis sequences-and-series cauchy-sequences
asked Jan 11 at 14:29
Cloud JRCloud JR
878517
$endgroup$
This question already has an answer here:
$f$ be a continuous function maps Cauchy into Cauchy. Is $f$ uniformly continuous?
1 answer
Does there exists a function which maps Cauchy sequence to Cauchy sequence but it isn't uniformly continuous?
we know uniformorly continuous function maps cauchy sequence to cauchy sequence.But my book doesn't say anything about converse of i think its false. but didn't find a counter example yet . please help!
This question already has an answer here:
$f$ be a continuous function maps Cauchy into Cauchy. Is $f$ uniformly continuous?
1 answer
real-analysis sequences-and-series cauchy-sequences
real-analysis sequences-and-series cauchy-sequences
asked Jan 11 at 14:29
Cloud JRCloud JR
878517
asked Jan 11 at 14:29
Cloud JRCloud JR
878517
asked Jan 11 at 14:29
Cloud JRCloud JR
878517
asked Jan 11 at 14:29
Cloud JRCloud JR
878517
asked Jan 11 at 14:29
Cloud JRCloud JR
878517
878517
marked as duplicate by Martin R, Community♦ Jan 11 at 14:34
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Martin R, Community♦ Jan 11 at 14:34
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
2
$begingroup$
$f : x mapsto x^2$ on $Bbb R$
$endgroup$
– Chinnapparaj R
Jan 11 at 14:29
$begingroup$
Functions that map Cauchy sequences to Cauchy sequences are called Cauchy-continuous functitons. It's a nice little exercise to show that Cauchy-continuity is stronger than continuity.
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– Theo Bendit
Jan 11 at 14:36
add a comment |
2
$begingroup$
$f : x mapsto x^2$ on $Bbb R$
$endgroup$
– Chinnapparaj R
Jan 11 at 14:29
$begingroup$
Functions that map Cauchy sequences to Cauchy sequences are called Cauchy-continuous functitons. It's a nice little exercise to show that Cauchy-continuity is stronger than continuity.
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– Theo Bendit
Jan 11 at 14:36
2
2
$begingroup$
$f : x mapsto x^2$ on $Bbb R$
$endgroup$
– Chinnapparaj R
Jan 11 at 14:29
$begingroup$
$f : x mapsto x^2$ on $Bbb R$
$endgroup$
– Chinnapparaj R
Jan 11 at 14:29
$begingroup$
Functions that map Cauchy sequences to Cauchy sequences are called Cauchy-continuous functitons. It's a nice little exercise to show that Cauchy-continuity is stronger than continuity.
$endgroup$
– Theo Bendit
Jan 11 at 14:36
$begingroup$
Functions that map Cauchy sequences to Cauchy sequences are called Cauchy-continuous functitons. It's a nice little exercise to show that Cauchy-continuity is stronger than continuity.
$endgroup$
– Theo Bendit
Jan 11 at 14:36
add a comment |
2 Answers
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Since, in $mathbb R$, a sequence is a Cauchy sequence if and only if it converges, what you are asking for is a non-uniformly continuous map from $mathbb R$ into itself which maps convergent sequences into convergent sequences. Any continuous but non-uniformly continuous map will do that. Take any polynomial function whose degree is at least $2$, for instance.
answered Jan 11 at 14:32
José Carlos SantosJosé Carlos Santos
156k22125227
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Every continuous map on $mathbb{R}$ sends a Cauchy sequence to a Cauchy sequence (because Cauchy sequences convergence).
So you can pick any continuous but non uniformly continuous map on $mathbb{R}$ as a counter-example. Say $f(x)=x^2$.
answered Jan 11 at 14:32
HelloHello
313
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add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Since, in $mathbb R$, a sequence is a Cauchy sequence if and only if it converges, what you are asking for is a non-uniformly continuous map from $mathbb R$ into itself which maps convergent sequences into convergent sequences. Any continuous but non-uniformly continuous map will do that. Take any polynomial function whose degree is at least $2$, for instance.
answered Jan 11 at 14:32
José Carlos SantosJosé Carlos Santos
156k22125227
$endgroup$
add a comment |
$begingroup$
Since, in $mathbb R$, a sequence is a Cauchy sequence if and only if it converges, what you are asking for is a non-uniformly continuous map from $mathbb R$ into itself which maps convergent sequences into convergent sequences. Any continuous but non-uniformly continuous map will do that. Take any polynomial function whose degree is at least $2$, for instance.
answered Jan 11 at 14:32
José Carlos SantosJosé Carlos Santos
156k22125227
$endgroup$
add a comment |
$begingroup$
Since, in $mathbb R$, a sequence is a Cauchy sequence if and only if it converges, what you are asking for is a non-uniformly continuous map from $mathbb R$ into itself which maps convergent sequences into convergent sequences. Any continuous but non-uniformly continuous map will do that. Take any polynomial function whose degree is at least $2$, for instance.
answered Jan 11 at 14:32
José Carlos SantosJosé Carlos Santos
156k22125227
$endgroup$
Since, in $mathbb R$, a sequence is a Cauchy sequence if and only if it converges, what you are asking for is a non-uniformly continuous map from $mathbb R$ into itself which maps convergent sequences into convergent sequences. Any continuous but non-uniformly continuous map will do that. Take any polynomial function whose degree is at least $2$, for instance.
answered Jan 11 at 14:32
José Carlos SantosJosé Carlos Santos
156k22125227
answered Jan 11 at 14:32
José Carlos SantosJosé Carlos Santos
156k22125227
answered Jan 11 at 14:32
José Carlos SantosJosé Carlos Santos
156k22125227
answered Jan 11 at 14:32
José Carlos SantosJosé Carlos Santos
156k22125227
156k22125227
add a comment |
add a comment |
$begingroup$
Every continuous map on $mathbb{R}$ sends a Cauchy sequence to a Cauchy sequence (because Cauchy sequences convergence).
So you can pick any continuous but non uniformly continuous map on $mathbb{R}$ as a counter-example. Say $f(x)=x^2$.
answered Jan 11 at 14:32
HelloHello
313
$endgroup$
add a comment |
$begingroup$
Every continuous map on $mathbb{R}$ sends a Cauchy sequence to a Cauchy sequence (because Cauchy sequences convergence).
So you can pick any continuous but non uniformly continuous map on $mathbb{R}$ as a counter-example. Say $f(x)=x^2$.
answered Jan 11 at 14:32
HelloHello
313
$endgroup$
add a comment |
$begingroup$
Every continuous map on $mathbb{R}$ sends a Cauchy sequence to a Cauchy sequence (because Cauchy sequences convergence).
So you can pick any continuous but non uniformly continuous map on $mathbb{R}$ as a counter-example. Say $f(x)=x^2$.
answered Jan 11 at 14:32
HelloHello
313
$endgroup$
Every continuous map on $mathbb{R}$ sends a Cauchy sequence to a Cauchy sequence (because Cauchy sequences convergence).
So you can pick any continuous but non uniformly continuous map on $mathbb{R}$ as a counter-example. Say $f(x)=x^2$.
answered Jan 11 at 14:32
HelloHello
313
answered Jan 11 at 14:32
HelloHello
313
answered Jan 11 at 14:32
HelloHello
313
answered Jan 11 at 14:32
HelloHello
313
313
add a comment |
add a comment |
2
$begingroup$
$f : x mapsto x^2$ on $Bbb R$
$endgroup$
– Chinnapparaj R
Jan 11 at 14:29
$begingroup$
Functions that map Cauchy sequences to Cauchy sequences are called Cauchy-continuous functitons. It's a nice little exercise to show that Cauchy-continuity is stronger than continuity.
$endgroup$
– Theo Bendit
Jan 11 at 14:36