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Does pointwise convergence to a continuous function in compact set imply uniform convergence? [duplicate]
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This question already has an answer here:
Does pointwise convergence of continuous functions on a compact set to a continuous limit imply uniform convergence on that set?
1 answer
Suppose $K$ is a compact set in $mathbb{R}$, $f_n$ is a sequence of functions such that $f_n$ converges pointwise to a continuous function $f$, does it imply $f_n$ converges uniformly to $f$ in $K$ ?
Edit:
(addition of Hypothesis)$f_n$ is continuous for each $n$
If it is not true, what example justifies it ?
real-analysis uniform-convergence
asked Jan 18 at 17:25
Vinay SipaniVinay Sipani
635
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marked as duplicate by Hans Lundmark, RRL real-analysis
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Jan 18 at 19:37
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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This question already has an answer here:
Does pointwise convergence of continuous functions on a compact set to a continuous limit imply uniform convergence on that set?
1 answer
Suppose $K$ is a compact set in $mathbb{R}$, $f_n$ is a sequence of functions such that $f_n$ converges pointwise to a continuous function $f$, does it imply $f_n$ converges uniformly to $f$ in $K$ ?
Edit:
(addition of Hypothesis)$f_n$ is continuous for each $n$
If it is not true, what example justifies it ?
real-analysis uniform-convergence
asked Jan 18 at 17:25
Vinay SipaniVinay Sipani
635
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marked as duplicate by Hans Lundmark, RRL real-analysis
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Jan 18 at 19:37
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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I believe this is the statement of Dini’s Theorem. In that case, yes.
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– Fede Poncio
Jan 18 at 17:29
1
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@FedePoncio I believe Dini's Theorem has a monotonicity assumption
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– angryavian
Jan 18 at 17:30
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Actually, Dini requires $f_n$ to be either increasing or decreasing. Without that condition I don’t think it holds
$endgroup$
– Fede Poncio
Jan 18 at 17:30
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@angryavian Yes, thank you
$endgroup$
– Fede Poncio
Jan 18 at 17:30
1
$begingroup$
Lokk here for various counterexamples showing all hypotheses of Dini's theorem are necessary.
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– saulspatz
Jan 18 at 17:44
|
show 4 more comments
$begingroup$
This question already has an answer here:
Does pointwise convergence of continuous functions on a compact set to a continuous limit imply uniform convergence on that set?
1 answer
Suppose $K$ is a compact set in $mathbb{R}$, $f_n$ is a sequence of functions such that $f_n$ converges pointwise to a continuous function $f$, does it imply $f_n$ converges uniformly to $f$ in $K$ ?
Edit:
(addition of Hypothesis)$f_n$ is continuous for each $n$
If it is not true, what example justifies it ?
real-analysis uniform-convergence
asked Jan 18 at 17:25
Vinay SipaniVinay Sipani
635
$endgroup$
This question already has an answer here:
Does pointwise convergence of continuous functions on a compact set to a continuous limit imply uniform convergence on that set?
1 answer
Suppose $K$ is a compact set in $mathbb{R}$, $f_n$ is a sequence of functions such that $f_n$ converges pointwise to a continuous function $f$, does it imply $f_n$ converges uniformly to $f$ in $K$ ?
Edit:
(addition of Hypothesis)$f_n$ is continuous for each $n$
If it is not true, what example justifies it ?
This question already has an answer here:
Does pointwise convergence of continuous functions on a compact set to a continuous limit imply uniform convergence on that set?
1 answer
real-analysis uniform-convergence
real-analysis uniform-convergence
asked Jan 18 at 17:25
Vinay SipaniVinay Sipani
635
asked Jan 18 at 17:25
Vinay SipaniVinay Sipani
635
edited Jan 18 at 17:36
Vinay Sipani
asked Jan 18 at 17:25
Vinay SipaniVinay Sipani
635
asked Jan 18 at 17:25
Vinay SipaniVinay Sipani
635
asked Jan 18 at 17:25
Vinay SipaniVinay Sipani
635
635
marked as duplicate by Hans Lundmark, RRL real-analysis
Users with the real-analysis badge can single-handedly close real-analysis questions as duplicates and reopen them as needed.
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Jan 18 at 19:37
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 Hans Lundmark, RRL real-analysis
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Jan 18 at 19:37
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.
$begingroup$
I believe this is the statement of Dini’s Theorem. In that case, yes.
$endgroup$
– Fede Poncio
Jan 18 at 17:29
1
$begingroup$
@FedePoncio I believe Dini's Theorem has a monotonicity assumption
$endgroup$
– angryavian
Jan 18 at 17:30
$begingroup$
Actually, Dini requires $f_n$ to be either increasing or decreasing. Without that condition I don’t think it holds
$endgroup$
– Fede Poncio
Jan 18 at 17:30
$begingroup$
@angryavian Yes, thank you
$endgroup$
– Fede Poncio
Jan 18 at 17:30
1
$begingroup$
Lokk here for various counterexamples showing all hypotheses of Dini's theorem are necessary.
$endgroup$
– saulspatz
Jan 18 at 17:44
|
show 4 more comments
$begingroup$
I believe this is the statement of Dini’s Theorem. In that case, yes.
$endgroup$
– Fede Poncio
Jan 18 at 17:29
1
$begingroup$
@FedePoncio I believe Dini's Theorem has a monotonicity assumption
$endgroup$
– angryavian
Jan 18 at 17:30
$begingroup$
Actually, Dini requires $f_n$ to be either increasing or decreasing. Without that condition I don’t think it holds
$endgroup$
– Fede Poncio
Jan 18 at 17:30
$begingroup$
@angryavian Yes, thank you
$endgroup$
– Fede Poncio
Jan 18 at 17:30
1
$begingroup$
Lokk here for various counterexamples showing all hypotheses of Dini's theorem are necessary.
$endgroup$
– saulspatz
Jan 18 at 17:44
$begingroup$
I believe this is the statement of Dini’s Theorem. In that case, yes.
$endgroup$
– Fede Poncio
Jan 18 at 17:29
$begingroup$
I believe this is the statement of Dini’s Theorem. In that case, yes.
$endgroup$
– Fede Poncio
Jan 18 at 17:29
1
1
$begingroup$
@FedePoncio I believe Dini's Theorem has a monotonicity assumption
$endgroup$
– angryavian
Jan 18 at 17:30
$begingroup$
@FedePoncio I believe Dini's Theorem has a monotonicity assumption
$endgroup$
– angryavian
Jan 18 at 17:30
$begingroup$
Actually, Dini requires $f_n$ to be either increasing or decreasing. Without that condition I don’t think it holds
$endgroup$
– Fede Poncio
Jan 18 at 17:30
$begingroup$
Actually, Dini requires $f_n$ to be either increasing or decreasing. Without that condition I don’t think it holds
$endgroup$
– Fede Poncio
Jan 18 at 17:30
$begingroup$
@angryavian Yes, thank you
$endgroup$
– Fede Poncio
Jan 18 at 17:30
$begingroup$
@angryavian Yes, thank you
$endgroup$
– Fede Poncio
Jan 18 at 17:30
1
1
$begingroup$
Lokk here for various counterexamples showing all hypotheses of Dini's theorem are necessary.
$endgroup$
– saulspatz
Jan 18 at 17:44
$begingroup$
Lokk here for various counterexamples showing all hypotheses of Dini's theorem are necessary.
$endgroup$
– saulspatz
Jan 18 at 17:44
|
show 4 more comments
2 Answers
2
active
oldest
votes
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No, if there is no condition that $f_n$ are continuous. Counterexample: $K = [0,1]$, $f_n(x)$ $=$ $0$ for $xneq1/n$, $1$ for $x = 1/n$, $f(x) equiv 0$.
EDIT: For continuous functions it is still not true. $K$ and $f$ as above, and $f_n(x) = 0$ for x < $1/2n$, $0$ for $x > 3/2n$, $1$ for $1/n$ and linear on intervals $[1/2n,1/n]$ and $[1/n,3/2n]$.
answered Jan 18 at 17:31
Jakub AndruszkiewiczJakub Andruszkiewicz
2116
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$begingroup$
Oh.. sorry. I missed to mention that $f_n$'s are continuous.
$endgroup$
– Vinay Sipani
Jan 18 at 17:33
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Thank you. That example works.
$endgroup$
– Vinay Sipani
Jan 18 at 17:42
add a comment |
$begingroup$
Let $X = [0, 2]$ and $$f_n(x) =begin{cases}
n^2x^2 − 2nx, 0 leq x leq 2/n\
0,&2/nlt x leq 2\
end{cases}$$
Clearly $f_n $ is continuous and converges pointwise to $0$, but the convergence is not uniform.
answered Jan 18 at 17:43
Thomas ShelbyThomas Shelby
3,7092525
$endgroup$
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
No, if there is no condition that $f_n$ are continuous. Counterexample: $K = [0,1]$, $f_n(x)$ $=$ $0$ for $xneq1/n$, $1$ for $x = 1/n$, $f(x) equiv 0$.
EDIT: For continuous functions it is still not true. $K$ and $f$ as above, and $f_n(x) = 0$ for x < $1/2n$, $0$ for $x > 3/2n$, $1$ for $1/n$ and linear on intervals $[1/2n,1/n]$ and $[1/n,3/2n]$.
answered Jan 18 at 17:31
Jakub AndruszkiewiczJakub Andruszkiewicz
2116
$endgroup$
$begingroup$
Oh.. sorry. I missed to mention that $f_n$'s are continuous.
$endgroup$
– Vinay Sipani
Jan 18 at 17:33
$begingroup$
Thank you. That example works.
$endgroup$
– Vinay Sipani
Jan 18 at 17:42
add a comment |
$begingroup$
No, if there is no condition that $f_n$ are continuous. Counterexample: $K = [0,1]$, $f_n(x)$ $=$ $0$ for $xneq1/n$, $1$ for $x = 1/n$, $f(x) equiv 0$.
EDIT: For continuous functions it is still not true. $K$ and $f$ as above, and $f_n(x) = 0$ for x < $1/2n$, $0$ for $x > 3/2n$, $1$ for $1/n$ and linear on intervals $[1/2n,1/n]$ and $[1/n,3/2n]$.
answered Jan 18 at 17:31
Jakub AndruszkiewiczJakub Andruszkiewicz
2116
$endgroup$
$begingroup$
Oh.. sorry. I missed to mention that $f_n$'s are continuous.
$endgroup$
– Vinay Sipani
Jan 18 at 17:33
$begingroup$
Thank you. That example works.
$endgroup$
– Vinay Sipani
Jan 18 at 17:42
add a comment |
$begingroup$
No, if there is no condition that $f_n$ are continuous. Counterexample: $K = [0,1]$, $f_n(x)$ $=$ $0$ for $xneq1/n$, $1$ for $x = 1/n$, $f(x) equiv 0$.
EDIT: For continuous functions it is still not true. $K$ and $f$ as above, and $f_n(x) = 0$ for x < $1/2n$, $0$ for $x > 3/2n$, $1$ for $1/n$ and linear on intervals $[1/2n,1/n]$ and $[1/n,3/2n]$.
answered Jan 18 at 17:31
Jakub AndruszkiewiczJakub Andruszkiewicz
2116
$endgroup$
No, if there is no condition that $f_n$ are continuous. Counterexample: $K = [0,1]$, $f_n(x)$ $=$ $0$ for $xneq1/n$, $1$ for $x = 1/n$, $f(x) equiv 0$.
EDIT: For continuous functions it is still not true. $K$ and $f$ as above, and $f_n(x) = 0$ for x < $1/2n$, $0$ for $x > 3/2n$, $1$ for $1/n$ and linear on intervals $[1/2n,1/n]$ and $[1/n,3/2n]$.
answered Jan 18 at 17:31
Jakub AndruszkiewiczJakub Andruszkiewicz
2116
edited Jan 18 at 17:35
answered Jan 18 at 17:31
Jakub AndruszkiewiczJakub Andruszkiewicz
2116
answered Jan 18 at 17:31
Jakub AndruszkiewiczJakub Andruszkiewicz
2116
answered Jan 18 at 17:31
Jakub AndruszkiewiczJakub Andruszkiewicz
2116
2116
$begingroup$
Oh.. sorry. I missed to mention that $f_n$'s are continuous.
$endgroup$
– Vinay Sipani
Jan 18 at 17:33
$begingroup$
Thank you. That example works.
$endgroup$
– Vinay Sipani
Jan 18 at 17:42
add a comment |
$begingroup$
Oh.. sorry. I missed to mention that $f_n$'s are continuous.
$endgroup$
– Vinay Sipani
Jan 18 at 17:33
$begingroup$
Thank you. That example works.
$endgroup$
– Vinay Sipani
Jan 18 at 17:42
$begingroup$
Oh.. sorry. I missed to mention that $f_n$'s are continuous.
$endgroup$
– Vinay Sipani
Jan 18 at 17:33
$begingroup$
Oh.. sorry. I missed to mention that $f_n$'s are continuous.
$endgroup$
– Vinay Sipani
Jan 18 at 17:33
$begingroup$
Thank you. That example works.
$endgroup$
– Vinay Sipani
Jan 18 at 17:42
$begingroup$
Thank you. That example works.
$endgroup$
– Vinay Sipani
Jan 18 at 17:42
add a comment |
$begingroup$
Let $X = [0, 2]$ and $$f_n(x) =begin{cases}
n^2x^2 − 2nx, 0 leq x leq 2/n\
0,&2/nlt x leq 2\
end{cases}$$
Clearly $f_n $ is continuous and converges pointwise to $0$, but the convergence is not uniform.
answered Jan 18 at 17:43
Thomas ShelbyThomas Shelby
3,7092525
$endgroup$
add a comment |
$begingroup$
Let $X = [0, 2]$ and $$f_n(x) =begin{cases}
n^2x^2 − 2nx, 0 leq x leq 2/n\
0,&2/nlt x leq 2\
end{cases}$$
Clearly $f_n $ is continuous and converges pointwise to $0$, but the convergence is not uniform.
answered Jan 18 at 17:43
Thomas ShelbyThomas Shelby
3,7092525
$endgroup$
add a comment |
$begingroup$
Let $X = [0, 2]$ and $$f_n(x) =begin{cases}
n^2x^2 − 2nx, 0 leq x leq 2/n\
0,&2/nlt x leq 2\
end{cases}$$
Clearly $f_n $ is continuous and converges pointwise to $0$, but the convergence is not uniform.
answered Jan 18 at 17:43
Thomas ShelbyThomas Shelby
3,7092525
$endgroup$
Let $X = [0, 2]$ and $$f_n(x) =begin{cases}
n^2x^2 − 2nx, 0 leq x leq 2/n\
0,&2/nlt x leq 2\
end{cases}$$
Clearly $f_n $ is continuous and converges pointwise to $0$, but the convergence is not uniform.
answered Jan 18 at 17:43
Thomas ShelbyThomas Shelby
3,7092525
answered Jan 18 at 17:43
Thomas ShelbyThomas Shelby
3,7092525
answered Jan 18 at 17:43
Thomas ShelbyThomas Shelby
3,7092525
answered Jan 18 at 17:43
Thomas ShelbyThomas Shelby
3,7092525
3,7092525
add a comment |
add a comment |
$begingroup$
I believe this is the statement of Dini’s Theorem. In that case, yes.
$endgroup$
– Fede Poncio
Jan 18 at 17:29
1
$begingroup$
@FedePoncio I believe Dini's Theorem has a monotonicity assumption
$endgroup$
– angryavian
Jan 18 at 17:30
$begingroup$
Actually, Dini requires $f_n$ to be either increasing or decreasing. Without that condition I don’t think it holds
$endgroup$
– Fede Poncio
Jan 18 at 17:30
$begingroup$
@angryavian Yes, thank you
$endgroup$
– Fede Poncio
Jan 18 at 17:30
1
$begingroup$
Lokk here for various counterexamples showing all hypotheses of Dini's theorem are necessary.
$endgroup$
– saulspatz
Jan 18 at 17:44