Mixing
Pointwise convergence with function valued on extended real line
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I just want to clarify if we allow $f(x)=infty$ when we define $f_n(x) to f(x)$ pointwise. Or $f_n to f$ would imply that $f(x) neq infty$ (this I assume is different from boundedness?).
Example: $f_n(x)=(1-|x|)n,-1le x le 1$ and $f=0$ otherwise. In this case, $f$ takes infinite values on $(-1,1)$. In this case, can we say $f_n$ converge to $f$ pointwise?
real-analysis
asked Aug 2 at 19:08
Daniel Li
486212
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down vote
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I just want to clarify if we allow $f(x)=infty$ when we define $f_n(x) to f(x)$ pointwise. Or $f_n to f$ would imply that $f(x) neq infty$ (this I assume is different from boundedness?).
Example: $f_n(x)=(1-|x|)n,-1le x le 1$ and $f=0$ otherwise. In this case, $f$ takes infinite values on $(-1,1)$. In this case, can we say $f_n$ converge to $f$ pointwise?
real-analysis
asked Aug 2 at 19:08
Daniel Li
486212
For downvote: I have made edits to clarify the question. Please let me know if it is still unclear.
– Daniel Li
Aug 2 at 19:25
This is question about definitions. There is no proof or counter-example involved. In some circumstances Mathematicians allow infinite values but in some other circumstances they don't.
– Kavi Rama Murthy
Aug 2 at 23:38
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I just want to clarify if we allow $f(x)=infty$ when we define $f_n(x) to f(x)$ pointwise. Or $f_n to f$ would imply that $f(x) neq infty$ (this I assume is different from boundedness?).
Example: $f_n(x)=(1-|x|)n,-1le x le 1$ and $f=0$ otherwise. In this case, $f$ takes infinite values on $(-1,1)$. In this case, can we say $f_n$ converge to $f$ pointwise?
real-analysis
asked Aug 2 at 19:08
Daniel Li
486212
I just want to clarify if we allow $f(x)=infty$ when we define $f_n(x) to f(x)$ pointwise. Or $f_n to f$ would imply that $f(x) neq infty$ (this I assume is different from boundedness?).
Example: $f_n(x)=(1-|x|)n,-1le x le 1$ and $f=0$ otherwise. In this case, $f$ takes infinite values on $(-1,1)$. In this case, can we say $f_n$ converge to $f$ pointwise?
real-analysis
asked Aug 2 at 19:08
Daniel Li
486212
edited Aug 2 at 19:23
asked Aug 2 at 19:08
Daniel Li
486212
asked Aug 2 at 19:08
Daniel Li
486212
asked Aug 2 at 19:08
Daniel Li
486212
486212
For downvote: I have made edits to clarify the question. Please let me know if it is still unclear.
– Daniel Li
Aug 2 at 19:25
This is question about definitions. There is no proof or counter-example involved. In some circumstances Mathematicians allow infinite values but in some other circumstances they don't.
– Kavi Rama Murthy
Aug 2 at 23:38
add a comment |Â
For downvote: I have made edits to clarify the question. Please let me know if it is still unclear.
– Daniel Li
Aug 2 at 19:25
This is question about definitions. There is no proof or counter-example involved. In some circumstances Mathematicians allow infinite values but in some other circumstances they don't.
– Kavi Rama Murthy
Aug 2 at 23:38
For downvote: I have made edits to clarify the question. Please let me know if it is still unclear.
– Daniel Li
Aug 2 at 19:25
For downvote: I have made edits to clarify the question. Please let me know if it is still unclear.
– Daniel Li
Aug 2 at 19:25
This is question about definitions. There is no proof or counter-example involved. In some circumstances Mathematicians allow infinite values but in some other circumstances they don't.
– Kavi Rama Murthy
Aug 2 at 23:38
This is question about definitions. There is no proof or counter-example involved. In some circumstances Mathematicians allow infinite values but in some other circumstances they don't.
– Kavi Rama Murthy
Aug 2 at 23:38
add a comment |Â
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For downvote: I have made edits to clarify the question. Please let me know if it is still unclear.
– Daniel Li
Aug 2 at 19:25
This is question about definitions. There is no proof or counter-example involved. In some circumstances Mathematicians allow infinite values but in some other circumstances they don't.
– Kavi Rama Murthy
Aug 2 at 23:38