<p>The answer to this interesting question is <em>no</em>, the <em>Implicit Function Theorem</em> is a <strong>local</strong> result, and cannot be used to state the existence of a global solution.</p>
<p>A simple, usual, counterexample for two variables, is the circumference.
Consider the unit circumference with center at <span class="math-container">$0$</span>:</p>
<p><span class="math-container">$$x^2+y^2=1$$</span></p>
<p>and consider  the implicit function:</p>
<p><span class="math-container">$$F(x,y)= x^2+y^2-1=0,\;\;\;\;\tag{1}$$</span>
with  <span class="math-container">$x\in (-1,1)$</span>, so that <span class="math-container">$y\neq0$</span>.</p>
<p>The derivative <span class="math-container">$\frac{\partial F}{\partial y}$</span> is everywhere different from zero in the domain, but there isn't a single, unique function of one variable <span class="math-container">$y=f(x)$</span>, that makes explicit <span class="math-container">$(1)$</span>, as the equation has two solutions:</p>
<p><span class="math-container">$$y=\sqrt {1-x^2}$$</span> and
<span class="math-container">$$y=-\sqrt {1-x^2}$$</span></p>
<p>that is we have a function for the part of the circumference above the <span class="math-container">$x$</span> axis and a different  function for the part below.
<span class="math-container">$$$$</span></p>
<p>The subject of <strong>conditions based on Jacobian to establish the existence of solutions</strong> of systems has a long story and is a complicated matter.</p>
<p>In the  same way as the implicit theorem function is intimately linked to the  <em>Inverse Function Theorem</em> (local invertibility), the existence of global solutions is linked to the <strong>problem of global inversion of a function</strong>, a complex mathematical subject, also intertwined with the history of mathematical economics.</p>
<p>Consider that an important result, the <em><strong>Gale-Nikaido theorem</strong></em> (1965), was stated in response to a wrong theorem stated by Paul Samuelson: Gale and Nikaido exhibited a counterexample to Samuelson's result and elaborated their theorem.</p>
<p>You can refer to</p>
<p><a href="https://link.springer.com/article/10.1007/BF01360282" rel="noreferrer">Gale, Nikaido, The Jacobian matrix and global univalence of mappings (1965</a></p>
<p>and</p>
<p><a href="https://books.google.it/books?hl=it&amp;lr=&amp;id=kuh7CwAAQBAJ&amp;oi=fnd&amp;pg=PR7&amp;dq=Gale,+D.+and+Nikaido,+H.+(1965)+The+Jacobian+Matrix+and+Global+Univalence+of+Mappings.&amp;ots=z26TyO5WFI&amp;sig=uZmr12ueduME84dEJ7kg5CifUPw#v=onepage&amp;q=Gale%2C%20D.%20and%20Nikaido%2C%20H.%20(1965)%20The%20Jacobian%20Matrix%20and%20Global%20Univalence%20of%20Mappings.&amp;f=false" rel="noreferrer">Parthasarathy, <em>On Global Univalence Theorems</em>, (1983), Springer</a></p>
<p>See also this discussion on MathStackExchange:</p>
<p><a href="https://math.stackexchange.com/questions/2319134/derivative-based-sufficient-conditions-for-contraction-mappings-for-multivariate">https://math.stackexchange.com/questions/2319134/derivative-based-sufficient-conditions-for-contraction-mappings-for-multivariate</a>.</p>
<p>A more general treatment of the global invertibility problem of a function, the milestone result of which is the <a href="https://en.wikipedia.org/wiki/Diffeomorphism" rel="noreferrer"><em><strong>Hadamard-Caccioppoli Theorem</strong></em></a>, can be found in:</p>
<p><a href="https://www.researchgate.net/publication/227103011_Global_inversion_of_functions_An_introduction" rel="noreferrer" title="1994">De Marco, Gorni, Zampieri, <em>Global Inversion of Functions. An introduction</em> (1994)</a></p>
<p>For an in-depth treatment of the Implicit Function Theorem,  its development and generalizations, you can see</p>
<p><a href="https://www.amazon.it/Implicit-Function-Theorem-History-Applications/dp/146145980X/ref=sr_1_fkmr0_1?__mk_it_IT=%C3%85M%C3%85%C5%BD%C3%95%C3%91&amp;crid=RP7NOA62JTQT&amp;dib=eyJ2IjoiMSJ9.kjdJYE2V2E3Hu4MZ4oDLvwlfd947aeLNwnNJVSJercl-AKtj9Ai48djeIk89OEuqdMR32JngEIoMxOOfMMbdM2OIyv847Hv9VPm31FP7Ot6vTsHx9B0yzdQ9mOCALXOv.J53nTlcaXvIcP340ILXQCPLuq6iw97yS7YH1X9k96eE&amp;dib_tag=se&amp;keywords=krantz+implicit+function&amp;qid=1722001274&amp;sprefix=krantz+implicit+function%2Caps%2C249&amp;sr=8-1-fkmr0" rel="noreferrer">Krantz, Parks, <em>The Implicit Function Theorem. History, Theory and Applications</em> (2014), Birkhäuser</a></p>

CC BY-SA 4.0
2024-07-26T16:26:06.800Z
Thank you for the counter example. I think I have a better understanding now. Non-zero Jacobians just ensures that there doesn't exist more than one y for same x satisfying G in the open ball around a point. But it can happen outside the ball. So in general unique function may not exist.
Dayne
CC BY-SA 4.0
2024-07-26T16:29:18.127Z
Also thanks for the references. Will check them slowly as they seem too heavy for me.
Dayne
CC BY-SA 4.0
2024-07-26T16:36:30.903Z
You pointed out a subtle, complicated problem, the papers I linked are really mathematically advanced, I mentioned them to give an idea of the complexity of the problem. If you are interested in the matter about the implicit function theorem, Hadamard-Caccioppoli theorem included,  the book of Krantz is more readable, not so advanced, and gives a comprehesive  idea.
BakerStreet
CC BY-SA 4.0
2024-07-26T14:52:57.360Z
2024-07-26T15:12:20.320Z
2024-07-26T15:12:20.320Z
BakerStreet
<p>Suppose I have two non-linear equations from a model:</p>
<p><span class="math-container">$$G_1(x,y,a,b,c)=0$$</span></p>
<p><span class="math-container">$$G_2(x,y,a,b,c)=0$$</span></p>
<p>The functions are defined on some set <span class="math-container">$V \subset \mathbb R^5$</span>. If we show that the determinant of the Jacobian is non-zero, i.e.: <span class="math-container">$\forall \,\, (x,y,a,b,c) \in V$</span></p>
<p><span class="math-container">$$\frac{\partial G_1}{\partial x}\cdot \frac{\partial G_2}{\partial y} - \frac{\partial G_2}{\partial x}\cdot \frac{\partial G_1}{\partial y} \ne 0$$</span></p>
<p>Is this sufficient to argue unique <span class="math-container">$C^1$</span> functions <span class="math-container">$x^*(a,b,c), y^*(a,b,c)$</span> exist satisfying the above solution?</p>
<p>Basically, I am asking that unlike the usual use of IFT for arguing local existence around some points in the domain. Can we generalize it for overall existence when in the specific case that the sufficient condition of non-zero jacobian holds throughout the domain?</p>
<p>Any reference to support the claim would be greatly helpful. If the answer is no then some intuition/example of why it would not hold? And what extra condition could ensure existence.</p>

<mathematical-economics>
Extending Implicit Function Theorem to Show Global Existence
Dayne
# Extending Implicit Function Theorem to Show Global Existence

(Asked by: username_0 on 2024-07-26)

Suppose I have two non-linear equations from a model:

$$G\_1(x,y,a,b,c)=0$$

$$G\_2(x,y,a,b,c)=0$$

The functions are defined on some set $V \subset \mathbb R^5$. If we show that the determinant of the Jacobian is non-zero, i.e.: $\forall \,\, (x,y,a,b,c) \in V$

$$\frac{\partial G\_1}{\partial x}\cdot \frac{\partial G\_2}{\partial y} - \frac{\partial G\_2}{\partial x}\cdot \frac{\partial G\_1}{\partial y} \ne 0$$

Is this sufficient to argue unique $C^1$ functions $x^\*(a,b,c), y^\*(a,b,c)$ exist satisfying the above solution?

Basically, I am asking that unlike the usual use of IFT for arguing local existence around some points in the domain. Can we generalize it for overall existence when in the specific case that the sufficient condition of non-zero jacobian holds throughout the domain?

Any reference to support the claim would be greatly helpful. If the answer is no then some intuition/example of why it would not hold? And what extra condition could ensure existence.

**Tags:** mathematical-economics


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## 1 answers

### Answer by username_1 on 2024-07-26. Score: 6 | Accepted answer

The answer to this interesting question is *no*, the *Implicit Function Theorem* is a **local** result, and cannot be used to state the existence of a global solution.

A simple, usual, counterexample for two variables, is the circumference.
Consider the unit circumference with center at $0$:

$$x^2+y^2=1$$

and consider the implicit function:

$$F(x,y)= x^2+y^2-1=0,\;\;\;\;\tag{1}$$
with $x\in (-1,1)$, so that $y\neq0$.

The derivative $\frac{\partial F}{\partial y}$ is everywhere different from zero in the domain, but there isn't a single, unique function of one variable $y=f(x)$, that makes explicit $(1)$, as the equation has two solutions:

$$y=\sqrt {1-x^2}$$ and
$$y=-\sqrt {1-x^2}$$

that is we have a function for the part of the circumference above the $x$ axis and a different function for the part below.
$$$$

The subject of **conditions based on Jacobian to establish the existence of solutions** of systems has a long story and is a complicated matter.

In the same way as the implicit theorem function is intimately linked to the *Inverse Function Theorem* (local invertibility), the existence of global solutions is linked to the **problem of global inversion of a function**, a complex mathematical subject, also intertwined with the history of mathematical economics.

Consider that an important result, the ***Gale-Nikaido theorem*** (1965), was stated in response to a wrong theorem stated by Paul Samuelson: Gale and Nikaido exhibited a counterexample to Samuelson's result and elaborated their theorem.

You can refer to

[Gale, Nikaido, The Jacobian matrix and global univalence of mappings (1965](https://link.springer.com/article/10.1007/BF01360282)

and

[Parthasarathy, *On Global Univalence Theorems*, (1983), Springer](https://books.google.it/books?hl=it&lr=&id=kuh7CwAAQBAJ&oi=fnd&pg=PR7&dq=Gale,+D.+and+Nikaido,+H.+(1965)+The+Jacobian+Matrix+and+Global+Univalence+of+Mappings.&ots=z26TyO5WFI&sig=uZmr12ueduME84dEJ7kg5CifUPw#v=onepage&q=Gale%2C%20D.%20and%20Nikaido%2C%20H.%20(1965)%20The%20Jacobian%20Matrix%20and%20Global%20Univalence%20of%20Mappings.&f=false)

See also this discussion on MathStackExchange:

<https://math.stackexchange.com/questions/2319134/derivative-based-sufficient-conditions-for-contraction-mappings-for-multivariate>.

A more general treatment of the global invertibility problem of a function, the milestone result of which is the [***Hadamard-Caccioppoli Theorem***](https://en.wikipedia.org/wiki/Diffeomorphism), can be found in:

[De Marco, Gorni, Zampieri, *Global Inversion of Functions. An introduction* (1994)](https://www.researchgate.net/publication/227103011_Global_inversion_of_functions_An_introduction "1994")

For an in-depth treatment of the Implicit Function Theorem, its development and generalizations, you can see

[Krantz, Parks, *The Implicit Function Theorem. History, Theory and Applications* (2014), Birkhäuser](https://www.amazon.it/Implicit-Function-Theorem-History-Applications/dp/146145980X/ref=sr_1_fkmr0_1?__mk_it_IT=%C3%85M%C3%85%C5%BD%C3%95%C3%91&crid=RP7NOA62JTQT&dib=eyJ2IjoiMSJ9.kjdJYE2V2E3Hu4MZ4oDLvwlfd947aeLNwnNJVSJercl-AKtj9Ai48djeIk89OEuqdMR32JngEIoMxOOfMMbdM2OIyv847Hv9VPm31FP7Ot6vTsHx9B0yzdQ9mOCALXOv.J53nTlcaXvIcP340ILXQCPLuq6iw97yS7YH1X9k96eE&dib_tag=se&keywords=krantz+implicit+function&qid=1722001274&sprefix=krantz+implicit+function%2Caps%2C249&sr=8-1-fkmr0)

> **Comments:**
> **username_0**: Thank you for the counter example. I think I have a better understanding now. Non-zero Jacobians just ensures that there doesn't exist more than one y for same x satisfying G in the open ball around a point. But it can happen outside the ball. So in general unique function may not exist.  
> **username_0**: Also thanks for the references. Will check them slowly as they seem too heavy for me.  
> **username_1**: You pointed out a subtle, complicated problem, the papers I linked are really mathematically advanced, I mentioned them to give an idea of the complexity of the problem. If you are interested in the matter about the implicit function theorem, Hadamard-Caccioppoli theorem included,  the book of Krantz is more readable, not so advanced, and gives a comprehesive  idea.  

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