Uniform Continuity of sum of a series of functions
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Let $f(x)=sum_{n=1}^{infty}f_n(x)$, where $$f_n(x)=begin{cases}n(x-n+frac1{n}), xin[n-frac1{n},n]\n(n+frac1{n}-x), xin[n,n+frac1{n}]\0, text{otherwise}end{cases}$$
Then, is $f(x)$ uniformly continuous? I think no. But, I am unable to justify through rigour. By looking at the series of functions, I think it is pointwise convergent to zero. But, I dont think it is uniformly convergent. Any hints? Thanks beforehand.
real-analysis uniform-convergence uniform-continuity
asked Dec 18 '18 at 17:04
vidyarthividyarthi
3,0731833
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add a comment |
$begingroup$
Let $f(x)=sum_{n=1}^{infty}f_n(x)$, where $$f_n(x)=begin{cases}n(x-n+frac1{n}), xin[n-frac1{n},n]\n(n+frac1{n}-x), xin[n,n+frac1{n}]\0, text{otherwise}end{cases}$$
Then, is $f(x)$ uniformly continuous? I think no. But, I am unable to justify through rigour. By looking at the series of functions, I think it is pointwise convergent to zero. But, I dont think it is uniformly convergent. Any hints? Thanks beforehand.
real-analysis uniform-convergence uniform-continuity
asked Dec 18 '18 at 17:04
vidyarthividyarthi
3,0731833
$endgroup$
add a comment |
$begingroup$
Let $f(x)=sum_{n=1}^{infty}f_n(x)$, where $$f_n(x)=begin{cases}n(x-n+frac1{n}), xin[n-frac1{n},n]\n(n+frac1{n}-x), xin[n,n+frac1{n}]\0, text{otherwise}end{cases}$$
Then, is $f(x)$ uniformly continuous? I think no. But, I am unable to justify through rigour. By looking at the series of functions, I think it is pointwise convergent to zero. But, I dont think it is uniformly convergent. Any hints? Thanks beforehand.
real-analysis uniform-convergence uniform-continuity
asked Dec 18 '18 at 17:04
vidyarthividyarthi
3,0731833
$endgroup$
Let $f(x)=sum_{n=1}^{infty}f_n(x)$, where $$f_n(x)=begin{cases}n(x-n+frac1{n}), xin[n-frac1{n},n]\n(n+frac1{n}-x), xin[n,n+frac1{n}]\0, text{otherwise}end{cases}$$
Then, is $f(x)$ uniformly continuous? I think no. But, I am unable to justify through rigour. By looking at the series of functions, I think it is pointwise convergent to zero. But, I dont think it is uniformly convergent. Any hints? Thanks beforehand.
real-analysis uniform-convergence uniform-continuity
real-analysis uniform-convergence uniform-continuity
asked Dec 18 '18 at 17:04
vidyarthividyarthi
3,0731833
asked Dec 18 '18 at 17:04
vidyarthividyarthi
3,0731833
asked Dec 18 '18 at 17:04
vidyarthividyarthi
3,0731833
asked Dec 18 '18 at 17:04
vidyarthividyarthi
3,0731833
asked Dec 18 '18 at 17:04
vidyarthividyarthi
3,0731833
3,0731833
add a comment |
add a comment |
1 Answer
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$begingroup$
Hints: for your first question, what can you say about $f(n)-f(n+1/n)$?
For your second question, what is $f(n+1)-sum_{1 leq k leq n}{f_k(n+1)}$?
edited Dec 19 '18 at 3:23
vidyarthi
3,0731833
answered Dec 18 '18 at 17:38
MindlackMindlack
4,910211
$endgroup$
$begingroup$
so since $f(n)-f(n+frac1{n})to1$ as $ntoinfty$, therefore $f$ is not uniformly continuous. But, is the series uniformly convergent? I am unable to decipher the difference $f(n+1)-sum_{1le kle n}f_k(n+1)$
$endgroup$
– vidyarthi
Dec 19 '18 at 3:33
$begingroup$
What is $f_k(n+1)$ if $k < n$? What is $f_{n+1}(n+1)$?
$endgroup$
– Mindlack
Dec 19 '18 at 17:03
add a comment |
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1 Answer
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active
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1 Answer
1
active
oldest
votes
active
oldest
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active
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votes
$begingroup$
Hints: for your first question, what can you say about $f(n)-f(n+1/n)$?
For your second question, what is $f(n+1)-sum_{1 leq k leq n}{f_k(n+1)}$?
edited Dec 19 '18 at 3:23
vidyarthi
3,0731833
answered Dec 18 '18 at 17:38
MindlackMindlack
4,910211
$endgroup$
$begingroup$
so since $f(n)-f(n+frac1{n})to1$ as $ntoinfty$, therefore $f$ is not uniformly continuous. But, is the series uniformly convergent? I am unable to decipher the difference $f(n+1)-sum_{1le kle n}f_k(n+1)$
$endgroup$
– vidyarthi
Dec 19 '18 at 3:33
$begingroup$
What is $f_k(n+1)$ if $k < n$? What is $f_{n+1}(n+1)$?
$endgroup$
– Mindlack
Dec 19 '18 at 17:03
add a comment |
$begingroup$
Hints: for your first question, what can you say about $f(n)-f(n+1/n)$?
For your second question, what is $f(n+1)-sum_{1 leq k leq n}{f_k(n+1)}$?
edited Dec 19 '18 at 3:23
vidyarthi
3,0731833
answered Dec 18 '18 at 17:38
MindlackMindlack
4,910211
$endgroup$
$begingroup$
so since $f(n)-f(n+frac1{n})to1$ as $ntoinfty$, therefore $f$ is not uniformly continuous. But, is the series uniformly convergent? I am unable to decipher the difference $f(n+1)-sum_{1le kle n}f_k(n+1)$
$endgroup$
– vidyarthi
Dec 19 '18 at 3:33
$begingroup$
What is $f_k(n+1)$ if $k < n$? What is $f_{n+1}(n+1)$?
$endgroup$
– Mindlack
Dec 19 '18 at 17:03
add a comment |
$begingroup$
Hints: for your first question, what can you say about $f(n)-f(n+1/n)$?
For your second question, what is $f(n+1)-sum_{1 leq k leq n}{f_k(n+1)}$?
edited Dec 19 '18 at 3:23
vidyarthi
3,0731833
answered Dec 18 '18 at 17:38
MindlackMindlack
4,910211
$endgroup$
Hints: for your first question, what can you say about $f(n)-f(n+1/n)$?
For your second question, what is $f(n+1)-sum_{1 leq k leq n}{f_k(n+1)}$?
edited Dec 19 '18 at 3:23
vidyarthi
3,0731833
answered Dec 18 '18 at 17:38
MindlackMindlack
4,910211
edited Dec 19 '18 at 3:23
vidyarthi
3,0731833
edited Dec 19 '18 at 3:23
vidyarthi
3,0731833
edited Dec 19 '18 at 3:23
vidyarthi
3,0731833
3,0731833
answered Dec 18 '18 at 17:38
MindlackMindlack
4,910211
answered Dec 18 '18 at 17:38
MindlackMindlack
4,910211
answered Dec 18 '18 at 17:38
MindlackMindlack
4,910211
4,910211
$begingroup$
so since $f(n)-f(n+frac1{n})to1$ as $ntoinfty$, therefore $f$ is not uniformly continuous. But, is the series uniformly convergent? I am unable to decipher the difference $f(n+1)-sum_{1le kle n}f_k(n+1)$
$endgroup$
– vidyarthi
Dec 19 '18 at 3:33
$begingroup$
What is $f_k(n+1)$ if $k < n$? What is $f_{n+1}(n+1)$?
$endgroup$
– Mindlack
Dec 19 '18 at 17:03
add a comment |
$begingroup$
so since $f(n)-f(n+frac1{n})to1$ as $ntoinfty$, therefore $f$ is not uniformly continuous. But, is the series uniformly convergent? I am unable to decipher the difference $f(n+1)-sum_{1le kle n}f_k(n+1)$
$endgroup$
– vidyarthi
Dec 19 '18 at 3:33
$begingroup$
What is $f_k(n+1)$ if $k < n$? What is $f_{n+1}(n+1)$?
$endgroup$
– Mindlack
Dec 19 '18 at 17:03
$begingroup$
so since $f(n)-f(n+frac1{n})to1$ as $ntoinfty$, therefore $f$ is not uniformly continuous. But, is the series uniformly convergent? I am unable to decipher the difference $f(n+1)-sum_{1le kle n}f_k(n+1)$
$endgroup$
– vidyarthi
Dec 19 '18 at 3:33
$begingroup$
so since $f(n)-f(n+frac1{n})to1$ as $ntoinfty$, therefore $f$ is not uniformly continuous. But, is the series uniformly convergent? I am unable to decipher the difference $f(n+1)-sum_{1le kle n}f_k(n+1)$
$endgroup$
– vidyarthi
Dec 19 '18 at 3:33
$begingroup$
What is $f_k(n+1)$ if $k < n$? What is $f_{n+1}(n+1)$?
$endgroup$
– Mindlack
Dec 19 '18 at 17:03
$begingroup$
What is $f_k(n+1)$ if $k < n$? What is $f_{n+1}(n+1)$?
$endgroup$
– Mindlack
Dec 19 '18 at 17:03
add a comment |
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