“Counterexample” for the Inverse function theoremApplication of the Inverse Function TheoremQuestion regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(Omega)$.Looking for a special kind of injective functionInverse Function Theorem and InjectivityFind all $(x,y,z) in mathbbR^3$ where $f(x,y,z)=(xy,xz,yz)$ is locally invertibleInverse Function Theorem and global inversesInverse function theorem local injectivity proofFunction satisfying $(Df(x)h,h) geq alpha(h,h), forall x,h in mathbbR^n$ has an inverse around every point?Jordan Regions and the Inverse Function TheoremHow is this not a proof of the Jacobian conjecture in the complex case?
Is Big Ben visible from the British museum?
Why is Drogon so much better in battle than Rhaegal and Viserion?
Cannot remove door knob -- totally inaccessible!
Why aren't satellites disintegrated even though they orbit earth within their Roche Limits?
Have there been any examples of re-usable rockets in the past?
Capital gains on stocks sold to take initial investment off the table
Why is so much ransomware breakable?
How to deal with the extreme reverberation in big cathedrals when playing the pipe organs?
Square spiral in Mathematica
How could it be that 80% of townspeople were farmers during the Edo period in Japan?
Roman Numerals Equation 2
Cycling to work - 30mile return
Why does Taylor’s series “work”?
How do Ctrl+C and Ctrl+V work?
A latin word for "area of interest"
Would a "ring language" be possible?
How was the blinking terminal cursor invented?
What dog breeds survive the apocalypse for generations?
Why are there five extra turns in tournament Magic?
Is there any deeper thematic meaning to the white horse that Arya finds in The Bells (S08E05)?
Why do galaxies collide?
Why does string strummed with finger sound different from the one strummed with pick?
Deleting the same lines from a list
Given 0s on Assignments with suspected and dismissed cheating?
“Counterexample” for the Inverse function theorem
Application of the Inverse Function TheoremQuestion regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(Omega)$.Looking for a special kind of injective functionInverse Function Theorem and InjectivityFind all $(x,y,z) in mathbbR^3$ where $f(x,y,z)=(xy,xz,yz)$ is locally invertibleInverse Function Theorem and global inversesInverse function theorem local injectivity proofFunction satisfying $(Df(x)h,h) geq alpha(h,h), forall x,h in mathbbR^n$ has an inverse around every point?Jordan Regions and the Inverse Function TheoremHow is this not a proof of the Jacobian conjecture in the complex case?
$begingroup$
In my class we stated the theorem as follows:
Let $OmegasubseteqmathbbR^n$ be an open set and $f:OmegatomathbbR^n$ a $mathscrC^1(Omega)$ function. If $|J_f(a)|ne0$ for some $ainOmega$ then there exists $delta>0$ such that $g:=fvert_B(a,delta)$ is injective and ...
This only is a sufficient condition, so is there any function whose jacobian has determinant $0$ at every point but still is injective? If the determinant only vanished on one single point something similar to $f(x)=x^3$ at $x=0$ in $mathbbR$ would do the trick, but if $|f'(x)|=0$ for every $xinOmegasubseteqmathbbR$ then $f$ is constant and not injective. Does the same hold in $mathbbR^n$?
Thanks
real-analysis examples-counterexamples inverse-function-theorem
asked 2 hours ago
PedroPedro
661212
$endgroup$
add a comment |
$begingroup$
In my class we stated the theorem as follows:
Let $OmegasubseteqmathbbR^n$ be an open set and $f:OmegatomathbbR^n$ a $mathscrC^1(Omega)$ function. If $|J_f(a)|ne0$ for some $ainOmega$ then there exists $delta>0$ such that $g:=fvert_B(a,delta)$ is injective and ...
This only is a sufficient condition, so is there any function whose jacobian has determinant $0$ at every point but still is injective? If the determinant only vanished on one single point something similar to $f(x)=x^3$ at $x=0$ in $mathbbR$ would do the trick, but if $|f'(x)|=0$ for every $xinOmegasubseteqmathbbR$ then $f$ is constant and not injective. Does the same hold in $mathbbR^n$?
Thanks
real-analysis examples-counterexamples inverse-function-theorem
asked 2 hours ago
PedroPedro
661212
$endgroup$
add a comment |
$begingroup$
In my class we stated the theorem as follows:
Let $OmegasubseteqmathbbR^n$ be an open set and $f:OmegatomathbbR^n$ a $mathscrC^1(Omega)$ function. If $|J_f(a)|ne0$ for some $ainOmega$ then there exists $delta>0$ such that $g:=fvert_B(a,delta)$ is injective and ...
This only is a sufficient condition, so is there any function whose jacobian has determinant $0$ at every point but still is injective? If the determinant only vanished on one single point something similar to $f(x)=x^3$ at $x=0$ in $mathbbR$ would do the trick, but if $|f'(x)|=0$ for every $xinOmegasubseteqmathbbR$ then $f$ is constant and not injective. Does the same hold in $mathbbR^n$?
Thanks
real-analysis examples-counterexamples inverse-function-theorem
asked 2 hours ago
PedroPedro
661212
$endgroup$
In my class we stated the theorem as follows:
Let $OmegasubseteqmathbbR^n$ be an open set and $f:OmegatomathbbR^n$ a $mathscrC^1(Omega)$ function. If $|J_f(a)|ne0$ for some $ainOmega$ then there exists $delta>0$ such that $g:=fvert_B(a,delta)$ is injective and ...
This only is a sufficient condition, so is there any function whose jacobian has determinant $0$ at every point but still is injective? If the determinant only vanished on one single point something similar to $f(x)=x^3$ at $x=0$ in $mathbbR$ would do the trick, but if $|f'(x)|=0$ for every $xinOmegasubseteqmathbbR$ then $f$ is constant and not injective. Does the same hold in $mathbbR^n$?
Thanks
real-analysis examples-counterexamples inverse-function-theorem
real-analysis examples-counterexamples inverse-function-theorem
asked 2 hours ago
PedroPedro
661212
asked 2 hours ago
PedroPedro
661212
asked 2 hours ago
PedroPedro
661212
asked 2 hours ago
PedroPedro
661212
asked 2 hours ago
PedroPedro
661212
661212
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Actually, this is not possible in $mathbbR^n$ either.
Indeed, if you have any $mathscrC^1$ injective function $f: Omega rightarrow mathbbR^n$, then $f$ is open and a homeomorphism on its image (invariance of domain : https://en.m.wikipedia.org/wiki/Invariance_of_domain ).
From Sard’s theorem (https://en.m.wikipedia.org/wiki/Sard%27s_theorem ), the set of critical values has null measure in $mathbbR^n$, thus has empty interior, thus the set of critical points has no interior as well.
answered 2 hours ago
MindlackMindlack
5,100312
$endgroup$
add a comment |
Your Answer
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3227626%2fcounterexample-for-the-inverse-function-theorem%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Actually, this is not possible in $mathbbR^n$ either.
Indeed, if you have any $mathscrC^1$ injective function $f: Omega rightarrow mathbbR^n$, then $f$ is open and a homeomorphism on its image (invariance of domain : https://en.m.wikipedia.org/wiki/Invariance_of_domain ).
From Sard’s theorem (https://en.m.wikipedia.org/wiki/Sard%27s_theorem ), the set of critical values has null measure in $mathbbR^n$, thus has empty interior, thus the set of critical points has no interior as well.
answered 2 hours ago
MindlackMindlack
5,100312
$endgroup$
add a comment |
$begingroup$
Actually, this is not possible in $mathbbR^n$ either.
Indeed, if you have any $mathscrC^1$ injective function $f: Omega rightarrow mathbbR^n$, then $f$ is open and a homeomorphism on its image (invariance of domain : https://en.m.wikipedia.org/wiki/Invariance_of_domain ).
From Sard’s theorem (https://en.m.wikipedia.org/wiki/Sard%27s_theorem ), the set of critical values has null measure in $mathbbR^n$, thus has empty interior, thus the set of critical points has no interior as well.
answered 2 hours ago
MindlackMindlack
5,100312
$endgroup$
add a comment |
$begingroup$
Actually, this is not possible in $mathbbR^n$ either.
Indeed, if you have any $mathscrC^1$ injective function $f: Omega rightarrow mathbbR^n$, then $f$ is open and a homeomorphism on its image (invariance of domain : https://en.m.wikipedia.org/wiki/Invariance_of_domain ).
From Sard’s theorem (https://en.m.wikipedia.org/wiki/Sard%27s_theorem ), the set of critical values has null measure in $mathbbR^n$, thus has empty interior, thus the set of critical points has no interior as well.
answered 2 hours ago
MindlackMindlack
5,100312
$endgroup$
Actually, this is not possible in $mathbbR^n$ either.
Indeed, if you have any $mathscrC^1$ injective function $f: Omega rightarrow mathbbR^n$, then $f$ is open and a homeomorphism on its image (invariance of domain : https://en.m.wikipedia.org/wiki/Invariance_of_domain ).
From Sard’s theorem (https://en.m.wikipedia.org/wiki/Sard%27s_theorem ), the set of critical values has null measure in $mathbbR^n$, thus has empty interior, thus the set of critical points has no interior as well.
answered 2 hours ago
MindlackMindlack
5,100312
answered 2 hours ago
MindlackMindlack
5,100312
answered 2 hours ago
MindlackMindlack
5,100312
answered 2 hours ago
MindlackMindlack
5,100312
5,100312
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3227626%2fcounterexample-for-the-inverse-function-theorem%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
