Composition of regular functions not necessarily regular The 2019 Stack Overflow Developer Survey Results Are InComposition of functions help (Injection and Surjection)When does function composition commute?Function composition proof - proof that function is injectiveGraphical interpretation of composition of functionsWhy is ( Injections,∘ ) not a group but (Injections_finite, ∘) one?A unit ball is not a regular surfaceGiving examples for real functions with specific parametersComposition of functions - propertiesWhy is the composition of a surjective and injective function neither surjective nor injective?Prove that if $f$ is a diffeomorphism than its differential $D_f$ is an isomorphism
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Composition of regular functions not necessarily regular
The 2019 Stack Overflow Developer Survey Results Are InComposition of functions help (Injection and Surjection)When does function composition commute?Function composition proof - proof that function is injectiveGraphical interpretation of composition of functionsWhy is ( Injections,∘ ) not a group but (Injections_finite, ∘) one?A unit ball is not a regular surfaceGiving examples for real functions with specific parametersComposition of functions - propertiesWhy is the composition of a surjective and injective function neither surjective nor injective?Prove that if $f$ is a diffeomorphism than its differential $D_f$ is an isomorphism
$begingroup$
I define regular function in the following way:
Let $f: U to R^n$ a $C^1$ function, where $U subset R^m$ is an open set. Then, $f$ is regular if for every $x in U$, $rank(D_f(x)) = min(m,n)$.
Is there an example of two regular functions, that their composition is not regular?
I somehow struggle to find such example. What I did got is that neither of them can be a diffeomorphism. So maybe we need to look at functions that are not injective or that their inverse doesn't exist.
calculus multivariable-calculus functions
asked Mar 23 at 0:18
Gabi GGabi G
553210
$endgroup$
add a comment |
$begingroup$
I define regular function in the following way:
Let $f: U to R^n$ a $C^1$ function, where $U subset R^m$ is an open set. Then, $f$ is regular if for every $x in U$, $rank(D_f(x)) = min(m,n)$.
Is there an example of two regular functions, that their composition is not regular?
I somehow struggle to find such example. What I did got is that neither of them can be a diffeomorphism. So maybe we need to look at functions that are not injective or that their inverse doesn't exist.
calculus multivariable-calculus functions
asked Mar 23 at 0:18
Gabi GGabi G
553210
$endgroup$
add a comment |
$begingroup$
I define regular function in the following way:
Let $f: U to R^n$ a $C^1$ function, where $U subset R^m$ is an open set. Then, $f$ is regular if for every $x in U$, $rank(D_f(x)) = min(m,n)$.
Is there an example of two regular functions, that their composition is not regular?
I somehow struggle to find such example. What I did got is that neither of them can be a diffeomorphism. So maybe we need to look at functions that are not injective or that their inverse doesn't exist.
calculus multivariable-calculus functions
asked Mar 23 at 0:18
Gabi GGabi G
553210
$endgroup$
I define regular function in the following way:
Let $f: U to R^n$ a $C^1$ function, where $U subset R^m$ is an open set. Then, $f$ is regular if for every $x in U$, $rank(D_f(x)) = min(m,n)$.
Is there an example of two regular functions, that their composition is not regular?
I somehow struggle to find such example. What I did got is that neither of them can be a diffeomorphism. So maybe we need to look at functions that are not injective or that their inverse doesn't exist.
calculus multivariable-calculus functions
calculus multivariable-calculus functions
asked Mar 23 at 0:18
Gabi GGabi G
553210
asked Mar 23 at 0:18
Gabi GGabi G
553210
asked Mar 23 at 0:18
Gabi GGabi G
553210
asked Mar 23 at 0:18
Gabi GGabi G
553210
asked Mar 23 at 0:18
Gabi GGabi G
553210
553210
add a comment |
add a comment |
1 Answer
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$begingroup$
You can already see the problem in linear functions. Consider functions $f: mathbbR to mathbbR^2$ and $g: mathbbR^2 to mathbbR$ given by $f(x) = (x,0)$ and $g(x,y) = y$ so that $g circ f(x) = 0$, even though both are regular.
Under the additional hypothesis that both maps are surjective, or that both maps are injective, the composition of linear regular maps will be regular. Think of how you'd generalize this to nonlinear differentiable functions.
answered Mar 23 at 0:31
jacobianjacobian
1567
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1 Answer
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1 Answer
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$begingroup$
You can already see the problem in linear functions. Consider functions $f: mathbbR to mathbbR^2$ and $g: mathbbR^2 to mathbbR$ given by $f(x) = (x,0)$ and $g(x,y) = y$ so that $g circ f(x) = 0$, even though both are regular.
Under the additional hypothesis that both maps are surjective, or that both maps are injective, the composition of linear regular maps will be regular. Think of how you'd generalize this to nonlinear differentiable functions.
answered Mar 23 at 0:31
jacobianjacobian
1567
$endgroup$
add a comment |
$begingroup$
You can already see the problem in linear functions. Consider functions $f: mathbbR to mathbbR^2$ and $g: mathbbR^2 to mathbbR$ given by $f(x) = (x,0)$ and $g(x,y) = y$ so that $g circ f(x) = 0$, even though both are regular.
Under the additional hypothesis that both maps are surjective, or that both maps are injective, the composition of linear regular maps will be regular. Think of how you'd generalize this to nonlinear differentiable functions.
answered Mar 23 at 0:31
jacobianjacobian
1567
$endgroup$
add a comment |
$begingroup$
You can already see the problem in linear functions. Consider functions $f: mathbbR to mathbbR^2$ and $g: mathbbR^2 to mathbbR$ given by $f(x) = (x,0)$ and $g(x,y) = y$ so that $g circ f(x) = 0$, even though both are regular.
Under the additional hypothesis that both maps are surjective, or that both maps are injective, the composition of linear regular maps will be regular. Think of how you'd generalize this to nonlinear differentiable functions.
answered Mar 23 at 0:31
jacobianjacobian
1567
$endgroup$
You can already see the problem in linear functions. Consider functions $f: mathbbR to mathbbR^2$ and $g: mathbbR^2 to mathbbR$ given by $f(x) = (x,0)$ and $g(x,y) = y$ so that $g circ f(x) = 0$, even though both are regular.
Under the additional hypothesis that both maps are surjective, or that both maps are injective, the composition of linear regular maps will be regular. Think of how you'd generalize this to nonlinear differentiable functions.
answered Mar 23 at 0:31
jacobianjacobian
1567
answered Mar 23 at 0:31
jacobianjacobian
1567
answered Mar 23 at 0:31
jacobianjacobian
1567
answered Mar 23 at 0:31
jacobianjacobian
1567
1567
add a comment |
add a comment |
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