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Is this function continuous and differentiable at $x=0$?
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I'm a new math learner, and this question really concerns me though it's merely a freshman level one.
Let $Xsubset mathbbR$, $X=0cup(cup_textn is odd[frac1n+1,frac1n])$, and let $f:Xrightarrow mathbbR$, $f(x):=x$.
Is this function continuous and differentiable at $x=0$? At least, I think 0 is a limit point to $X$.
Thanks in advance for whoever can help me.
Cheers!~
analysis derivatives
asked yesterday
Collin Ren
63
add a comment |Â
up vote
1
down vote
favorite
I'm a new math learner, and this question really concerns me though it's merely a freshman level one.
Let $Xsubset mathbbR$, $X=0cup(cup_textn is odd[frac1n+1,frac1n])$, and let $f:Xrightarrow mathbbR$, $f(x):=x$.
Is this function continuous and differentiable at $x=0$? At least, I think 0 is a limit point to $X$.
Thanks in advance for whoever can help me.
Cheers!~
analysis derivatives
asked yesterday
Collin Ren
63
2
The standard definition of differentiability requires $x=0$ to be an interior point of $X$. However, it's not the case here.
– Jacky Chong
yesterday
1
Do you mean, X has to be such that there exists a $delta>0$: all $|x|<delta$ are in set X? And only then can we talk about both continuity and differentiability at 0, right?
– Collin Ren
yesterday
1
You don't need that for continuity, but you do need that for differentiability
– Kenny Lau
yesterday
Thank you very much! @KennyLau.
– Collin Ren
yesterday
@KennyLau if you could post your comment as answer, we can cross another question from the unanswered question queue.
– onurcanbektas
yesterday
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I'm a new math learner, and this question really concerns me though it's merely a freshman level one.
Let $Xsubset mathbbR$, $X=0cup(cup_textn is odd[frac1n+1,frac1n])$, and let $f:Xrightarrow mathbbR$, $f(x):=x$.
Is this function continuous and differentiable at $x=0$? At least, I think 0 is a limit point to $X$.
Thanks in advance for whoever can help me.
Cheers!~
analysis derivatives
asked yesterday
Collin Ren
63
I'm a new math learner, and this question really concerns me though it's merely a freshman level one.
Let $Xsubset mathbbR$, $X=0cup(cup_textn is odd[frac1n+1,frac1n])$, and let $f:Xrightarrow mathbbR$, $f(x):=x$.
Is this function continuous and differentiable at $x=0$? At least, I think 0 is a limit point to $X$.
Thanks in advance for whoever can help me.
Cheers!~
analysis derivatives
asked yesterday
Collin Ren
63
edited yesterday
asked yesterday
Collin Ren
63
asked yesterday
Collin Ren
63
asked yesterday
Collin Ren
63
63
2
The standard definition of differentiability requires $x=0$ to be an interior point of $X$. However, it's not the case here.
– Jacky Chong
yesterday
1
Do you mean, X has to be such that there exists a $delta>0$: all $|x|<delta$ are in set X? And only then can we talk about both continuity and differentiability at 0, right?
– Collin Ren
yesterday
1
You don't need that for continuity, but you do need that for differentiability
– Kenny Lau
yesterday
Thank you very much! @KennyLau.
– Collin Ren
yesterday
@KennyLau if you could post your comment as answer, we can cross another question from the unanswered question queue.
– onurcanbektas
yesterday
add a comment |Â
2
The standard definition of differentiability requires $x=0$ to be an interior point of $X$. However, it's not the case here.
– Jacky Chong
yesterday
1
Do you mean, X has to be such that there exists a $delta>0$: all $|x|<delta$ are in set X? And only then can we talk about both continuity and differentiability at 0, right?
– Collin Ren
yesterday
1
You don't need that for continuity, but you do need that for differentiability
– Kenny Lau
yesterday
Thank you very much! @KennyLau.
– Collin Ren
yesterday
@KennyLau if you could post your comment as answer, we can cross another question from the unanswered question queue.
– onurcanbektas
yesterday
2
2
The standard definition of differentiability requires $x=0$ to be an interior point of $X$. However, it's not the case here.
– Jacky Chong
yesterday
The standard definition of differentiability requires $x=0$ to be an interior point of $X$. However, it's not the case here.
– Jacky Chong
yesterday
1
1
Do you mean, X has to be such that there exists a $delta>0$: all $|x|<delta$ are in set X? And only then can we talk about both continuity and differentiability at 0, right?
– Collin Ren
yesterday
Do you mean, X has to be such that there exists a $delta>0$: all $|x|<delta$ are in set X? And only then can we talk about both continuity and differentiability at 0, right?
– Collin Ren
yesterday
1
1
You don't need that for continuity, but you do need that for differentiability
– Kenny Lau
yesterday
You don't need that for continuity, but you do need that for differentiability
– Kenny Lau
yesterday
Thank you very much! @KennyLau.
– Collin Ren
yesterday
Thank you very much! @KennyLau.
– Collin Ren
yesterday
@KennyLau if you could post your comment as answer, we can cross another question from the unanswered question queue.
– onurcanbektas
yesterday
@KennyLau if you could post your comment as answer, we can cross another question from the unanswered question queue.
– onurcanbektas
yesterday
add a comment |Â
1 Answer
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up vote
1
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This is more a question of the details of your definitions have been set up.
Intuitively, I would be comfortable with calling the function continuous and differentiable as a function $Xtomathbb R$, but probably not as a partial function $mathbb Rtomathbb R$.
Further, I would adopt a definition of at least "continuous" that made this true. Continuity at $x_0$ makes sense to me as long as $x_0$ is a point of the domain that's not isolated, and it's easy to say that in a definition, by giving the domain of the function the subspace topology.
For "differentiable", I'm less sure that it would be worth it to figure out a definition that is general enough to match my intuition here.
The overall points here are that:
- the details of definitions are a matter of choice, striking a balance between ease of use and being general enough to apply to the case you're interested in,
- not all authors/texts agree on these details.
- if you extend the definitions to cover these cases you need to at least consider whether the results you're using still hold -- by checking whether the source of those results use similarly broad definitions, or alternatively by checking whether you can adapt their proofs nevertheless,
- in most practical cases such results will apply. But that doesn't mean you don't need to check!
answered yesterday
Henning Makholm
225k16289515
Thank you very much sir @HenningMakholm, great point of view. Now I see.
– Collin Ren
yesterday
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
This is more a question of the details of your definitions have been set up.
Intuitively, I would be comfortable with calling the function continuous and differentiable as a function $Xtomathbb R$, but probably not as a partial function $mathbb Rtomathbb R$.
Further, I would adopt a definition of at least "continuous" that made this true. Continuity at $x_0$ makes sense to me as long as $x_0$ is a point of the domain that's not isolated, and it's easy to say that in a definition, by giving the domain of the function the subspace topology.
For "differentiable", I'm less sure that it would be worth it to figure out a definition that is general enough to match my intuition here.
The overall points here are that:
- the details of definitions are a matter of choice, striking a balance between ease of use and being general enough to apply to the case you're interested in,
- not all authors/texts agree on these details.
- if you extend the definitions to cover these cases you need to at least consider whether the results you're using still hold -- by checking whether the source of those results use similarly broad definitions, or alternatively by checking whether you can adapt their proofs nevertheless,
- in most practical cases such results will apply. But that doesn't mean you don't need to check!
answered yesterday
Henning Makholm
225k16289515
Thank you very much sir @HenningMakholm, great point of view. Now I see.
– Collin Ren
yesterday
add a comment |Â
up vote
1
down vote
This is more a question of the details of your definitions have been set up.
Intuitively, I would be comfortable with calling the function continuous and differentiable as a function $Xtomathbb R$, but probably not as a partial function $mathbb Rtomathbb R$.
Further, I would adopt a definition of at least "continuous" that made this true. Continuity at $x_0$ makes sense to me as long as $x_0$ is a point of the domain that's not isolated, and it's easy to say that in a definition, by giving the domain of the function the subspace topology.
For "differentiable", I'm less sure that it would be worth it to figure out a definition that is general enough to match my intuition here.
The overall points here are that:
- the details of definitions are a matter of choice, striking a balance between ease of use and being general enough to apply to the case you're interested in,
- not all authors/texts agree on these details.
- if you extend the definitions to cover these cases you need to at least consider whether the results you're using still hold -- by checking whether the source of those results use similarly broad definitions, or alternatively by checking whether you can adapt their proofs nevertheless,
- in most practical cases such results will apply. But that doesn't mean you don't need to check!
answered yesterday
Henning Makholm
225k16289515
Thank you very much sir @HenningMakholm, great point of view. Now I see.
– Collin Ren
yesterday
add a comment |Â
up vote
1
down vote
up vote
1
down vote
This is more a question of the details of your definitions have been set up.
Intuitively, I would be comfortable with calling the function continuous and differentiable as a function $Xtomathbb R$, but probably not as a partial function $mathbb Rtomathbb R$.
Further, I would adopt a definition of at least "continuous" that made this true. Continuity at $x_0$ makes sense to me as long as $x_0$ is a point of the domain that's not isolated, and it's easy to say that in a definition, by giving the domain of the function the subspace topology.
For "differentiable", I'm less sure that it would be worth it to figure out a definition that is general enough to match my intuition here.
The overall points here are that:
- the details of definitions are a matter of choice, striking a balance between ease of use and being general enough to apply to the case you're interested in,
- not all authors/texts agree on these details.
- if you extend the definitions to cover these cases you need to at least consider whether the results you're using still hold -- by checking whether the source of those results use similarly broad definitions, or alternatively by checking whether you can adapt their proofs nevertheless,
- in most practical cases such results will apply. But that doesn't mean you don't need to check!
answered yesterday
Henning Makholm
225k16289515
This is more a question of the details of your definitions have been set up.
Intuitively, I would be comfortable with calling the function continuous and differentiable as a function $Xtomathbb R$, but probably not as a partial function $mathbb Rtomathbb R$.
Further, I would adopt a definition of at least "continuous" that made this true. Continuity at $x_0$ makes sense to me as long as $x_0$ is a point of the domain that's not isolated, and it's easy to say that in a definition, by giving the domain of the function the subspace topology.
For "differentiable", I'm less sure that it would be worth it to figure out a definition that is general enough to match my intuition here.
The overall points here are that:
- the details of definitions are a matter of choice, striking a balance between ease of use and being general enough to apply to the case you're interested in,
- not all authors/texts agree on these details.
- if you extend the definitions to cover these cases you need to at least consider whether the results you're using still hold -- by checking whether the source of those results use similarly broad definitions, or alternatively by checking whether you can adapt their proofs nevertheless,
- in most practical cases such results will apply. But that doesn't mean you don't need to check!
answered yesterday
Henning Makholm
225k16289515
answered yesterday
Henning Makholm
225k16289515
answered yesterday
Henning Makholm
225k16289515
answered yesterday
Henning Makholm
225k16289515
225k16289515
Thank you very much sir @HenningMakholm, great point of view. Now I see.
– Collin Ren
yesterday
add a comment |Â
Thank you very much sir @HenningMakholm, great point of view. Now I see.
– Collin Ren
yesterday
Thank you very much sir @HenningMakholm, great point of view. Now I see.
– Collin Ren
yesterday
Thank you very much sir @HenningMakholm, great point of view. Now I see.
– Collin Ren
yesterday
add a comment |Â
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2
The standard definition of differentiability requires $x=0$ to be an interior point of $X$. However, it's not the case here.
– Jacky Chong
yesterday
1
Do you mean, X has to be such that there exists a $delta>0$: all $|x|<delta$ are in set X? And only then can we talk about both continuity and differentiability at 0, right?
– Collin Ren
yesterday
1
You don't need that for continuity, but you do need that for differentiability
– Kenny Lau
yesterday
Thank you very much! @KennyLau.
– Collin Ren
yesterday
@KennyLau if you could post your comment as answer, we can cross another question from the unanswered question queue.
– onurcanbektas
yesterday