Mixing
Every continuous open mapping from $mathbb{R}$ into $mathbb{R}$ is monotonic
$begingroup$
Consider the image of an open set $(a,b)$ under the open and continuous mapping $f$. We show, $f$ cannot have any extremum in $(a,b)$.
We know, connected sets are mapped to connected sets under a continuous map. Hence, $f[(a,b)]=(c,d)$ is connected (and open). Suppose, at $xi in (a,b)$, $f(xi)=sup_{(a,b)} f=M$ (or $inf_{(a,b)}f=m$). [Being a continuous function, $f$ must attain its supremum/infimum.]
Hence, the image of the set $(a,b)$ under $f$ becomes $(c,M]$ or $[m,d)$, which is a contradiction of the fact that $f$ maps open sets to open sets. Therefore, in any open interval, the function cannot attain glb/lub at an interior point. So, we conclude that the inf and sup are at the end points, i.e. $sup_{[a,b]}f=f(a)$ or $f(b)$.
Hence the theorem.
Is the proof valid? I am aware of the duplicates. I just want this method verified.
real-analysis proof-verification continuity monotone-functions open-map
asked Jan 28 at 17:17
Subhasis BiswasSubhasis Biswas
512412
$endgroup$
|
show 2 more comments
$begingroup$
Consider the image of an open set $(a,b)$ under the open and continuous mapping $f$. We show, $f$ cannot have any extremum in $(a,b)$.
We know, connected sets are mapped to connected sets under a continuous map. Hence, $f[(a,b)]=(c,d)$ is connected (and open). Suppose, at $xi in (a,b)$, $f(xi)=sup_{(a,b)} f=M$ (or $inf_{(a,b)}f=m$). [Being a continuous function, $f$ must attain its supremum/infimum.]
Hence, the image of the set $(a,b)$ under $f$ becomes $(c,M]$ or $[m,d)$, which is a contradiction of the fact that $f$ maps open sets to open sets. Therefore, in any open interval, the function cannot attain glb/lub at an interior point. So, we conclude that the inf and sup are at the end points, i.e. $sup_{[a,b]}f=f(a)$ or $f(b)$.
Hence the theorem.
Is the proof valid? I am aware of the duplicates. I just want this method verified.
real-analysis proof-verification continuity monotone-functions open-map
asked Jan 28 at 17:17
Subhasis BiswasSubhasis Biswas
512412
$endgroup$
$begingroup$
Possible duplicate of Every continuous open mapping of $mathbb{R}$ into $mathbb{R}$ is monotonic
$endgroup$
– Mees de Vries
Jan 28 at 17:21
$begingroup$
i am going to take down my previous question
$endgroup$
– Subhasis Biswas
Jan 28 at 17:23
$begingroup$
If the question is essentially the same, you just want to change the presentation, it makes more sense to edit the old question, rather than delete and then immediately re-ask.
$endgroup$
– Mees de Vries
Jan 28 at 17:23
$begingroup$
Thank you! will keep that in mind.
$endgroup$
– Subhasis Biswas
Jan 28 at 17:24
$begingroup$
Your proof is not correct, first because the extreme value theorem doesn't apply on an open interval, and second because such "global" reasoning doesn't invalidate the possibility of a function whose global extrema on $[a,b]$ are at the endpoints but nevertheless has some local extremum.
$endgroup$
– Ian
Jan 28 at 17:25
|
show 2 more comments
$begingroup$
Consider the image of an open set $(a,b)$ under the open and continuous mapping $f$. We show, $f$ cannot have any extremum in $(a,b)$.
We know, connected sets are mapped to connected sets under a continuous map. Hence, $f[(a,b)]=(c,d)$ is connected (and open). Suppose, at $xi in (a,b)$, $f(xi)=sup_{(a,b)} f=M$ (or $inf_{(a,b)}f=m$). [Being a continuous function, $f$ must attain its supremum/infimum.]
Hence, the image of the set $(a,b)$ under $f$ becomes $(c,M]$ or $[m,d)$, which is a contradiction of the fact that $f$ maps open sets to open sets. Therefore, in any open interval, the function cannot attain glb/lub at an interior point. So, we conclude that the inf and sup are at the end points, i.e. $sup_{[a,b]}f=f(a)$ or $f(b)$.
Hence the theorem.
Is the proof valid? I am aware of the duplicates. I just want this method verified.
real-analysis proof-verification continuity monotone-functions open-map
asked Jan 28 at 17:17
Subhasis BiswasSubhasis Biswas
512412
$endgroup$
Consider the image of an open set $(a,b)$ under the open and continuous mapping $f$. We show, $f$ cannot have any extremum in $(a,b)$.
We know, connected sets are mapped to connected sets under a continuous map. Hence, $f[(a,b)]=(c,d)$ is connected (and open). Suppose, at $xi in (a,b)$, $f(xi)=sup_{(a,b)} f=M$ (or $inf_{(a,b)}f=m$). [Being a continuous function, $f$ must attain its supremum/infimum.]
Hence, the image of the set $(a,b)$ under $f$ becomes $(c,M]$ or $[m,d)$, which is a contradiction of the fact that $f$ maps open sets to open sets. Therefore, in any open interval, the function cannot attain glb/lub at an interior point. So, we conclude that the inf and sup are at the end points, i.e. $sup_{[a,b]}f=f(a)$ or $f(b)$.
Hence the theorem.
Is the proof valid? I am aware of the duplicates. I just want this method verified.
real-analysis proof-verification continuity monotone-functions open-map
real-analysis proof-verification continuity monotone-functions open-map
asked Jan 28 at 17:17
Subhasis BiswasSubhasis Biswas
512412
asked Jan 28 at 17:17
Subhasis BiswasSubhasis Biswas
512412
edited Jan 28 at 17:35
Subhasis Biswas
asked Jan 28 at 17:17
Subhasis BiswasSubhasis Biswas
512412
asked Jan 28 at 17:17
Subhasis BiswasSubhasis Biswas
512412
asked Jan 28 at 17:17
Subhasis BiswasSubhasis Biswas
512412
512412
$begingroup$
Possible duplicate of Every continuous open mapping of $mathbb{R}$ into $mathbb{R}$ is monotonic
$endgroup$
– Mees de Vries
Jan 28 at 17:21
$begingroup$
i am going to take down my previous question
$endgroup$
– Subhasis Biswas
Jan 28 at 17:23
$begingroup$
If the question is essentially the same, you just want to change the presentation, it makes more sense to edit the old question, rather than delete and then immediately re-ask.
$endgroup$
– Mees de Vries
Jan 28 at 17:23
$begingroup$
Thank you! will keep that in mind.
$endgroup$
– Subhasis Biswas
Jan 28 at 17:24
$begingroup$
Your proof is not correct, first because the extreme value theorem doesn't apply on an open interval, and second because such "global" reasoning doesn't invalidate the possibility of a function whose global extrema on $[a,b]$ are at the endpoints but nevertheless has some local extremum.
$endgroup$
– Ian
Jan 28 at 17:25
|
show 2 more comments
$begingroup$
Possible duplicate of Every continuous open mapping of $mathbb{R}$ into $mathbb{R}$ is monotonic
$endgroup$
– Mees de Vries
Jan 28 at 17:21
$begingroup$
i am going to take down my previous question
$endgroup$
– Subhasis Biswas
Jan 28 at 17:23
$begingroup$
If the question is essentially the same, you just want to change the presentation, it makes more sense to edit the old question, rather than delete and then immediately re-ask.
$endgroup$
– Mees de Vries
Jan 28 at 17:23
$begingroup$
Thank you! will keep that in mind.
$endgroup$
– Subhasis Biswas
Jan 28 at 17:24
$begingroup$
Your proof is not correct, first because the extreme value theorem doesn't apply on an open interval, and second because such "global" reasoning doesn't invalidate the possibility of a function whose global extrema on $[a,b]$ are at the endpoints but nevertheless has some local extremum.
$endgroup$
– Ian
Jan 28 at 17:25
$begingroup$
Possible duplicate of Every continuous open mapping of $mathbb{R}$ into $mathbb{R}$ is monotonic
$endgroup$
– Mees de Vries
Jan 28 at 17:21
$begingroup$
Possible duplicate of Every continuous open mapping of $mathbb{R}$ into $mathbb{R}$ is monotonic
$endgroup$
– Mees de Vries
Jan 28 at 17:21
$begingroup$
i am going to take down my previous question
$endgroup$
– Subhasis Biswas
Jan 28 at 17:23
$begingroup$
i am going to take down my previous question
$endgroup$
– Subhasis Biswas
Jan 28 at 17:23
$begingroup$
If the question is essentially the same, you just want to change the presentation, it makes more sense to edit the old question, rather than delete and then immediately re-ask.
$endgroup$
– Mees de Vries
Jan 28 at 17:23
$begingroup$
If the question is essentially the same, you just want to change the presentation, it makes more sense to edit the old question, rather than delete and then immediately re-ask.
$endgroup$
– Mees de Vries
Jan 28 at 17:23
$begingroup$
Thank you! will keep that in mind.
$endgroup$
– Subhasis Biswas
Jan 28 at 17:24
$begingroup$
Thank you! will keep that in mind.
$endgroup$
– Subhasis Biswas
Jan 28 at 17:24
$begingroup$
Your proof is not correct, first because the extreme value theorem doesn't apply on an open interval, and second because such "global" reasoning doesn't invalidate the possibility of a function whose global extrema on $[a,b]$ are at the endpoints but nevertheless has some local extremum.
$endgroup$
– Ian
Jan 28 at 17:25
$begingroup$
Your proof is not correct, first because the extreme value theorem doesn't apply on an open interval, and second because such "global" reasoning doesn't invalidate the possibility of a function whose global extrema on $[a,b]$ are at the endpoints but nevertheless has some local extremum.
$endgroup$
– Ian
Jan 28 at 17:25
|
show 2 more comments
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$begingroup$
Possible duplicate of Every continuous open mapping of $mathbb{R}$ into $mathbb{R}$ is monotonic
$endgroup$
– Mees de Vries
Jan 28 at 17:21
$begingroup$
i am going to take down my previous question
$endgroup$
– Subhasis Biswas
Jan 28 at 17:23
$begingroup$
If the question is essentially the same, you just want to change the presentation, it makes more sense to edit the old question, rather than delete and then immediately re-ask.
$endgroup$
– Mees de Vries
Jan 28 at 17:23
$begingroup$
Thank you! will keep that in mind.
$endgroup$
– Subhasis Biswas
Jan 28 at 17:24
$begingroup$
Your proof is not correct, first because the extreme value theorem doesn't apply on an open interval, and second because such "global" reasoning doesn't invalidate the possibility of a function whose global extrema on $[a,b]$ are at the endpoints but nevertheless has some local extremum.
$endgroup$
– Ian
Jan 28 at 17:25