Questions tagged [inverse-function-theorem]

Recommended to be used when the Inverse function theorem is being employed in a question, and also for those users that need help understanding it.

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Let $f:\mathbb{R^2} \to \mathbb{R^2}$ whose derivative is nonsingular.let D denote the open unit ball $\{v:|v|<1\}$. show that $D \subset f(D)$

let $f:\mathbb{R^2} \to \mathbb{R^2}$ smooth function whose derivative at every point is nonsingular. suppose that $f(O)=O$ and for all $v \in R^2$ with $|v|=1$ , $|f(v)| \geq 1$. let D denote the ...
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Preimage of a 0-measure set by $f:R^n\to R^m$ with $m<n$ and continuous Jacobian everywhere of rank $m$ has measure 0

Consider $f:R^n\to R^m$, with continuous Jacobian $J(f)(x)$ of rank $m$ (we assume $n>m$) for all $x\in R^n$. Now let $E\subset R^m$ be a set of Lebesgue measure 0. Is it always true that $f^{-1}(E)...
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Parametric inverse mapping theorem

Let $M$ and $N$ be smooth manifolds. I was reading a proof of a result in which we have a map $F:[0,\infty[\times M\rightarrow N\times N$ with the property that $F(0)\in \triangle$, where $\triangle$ ...
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Solution of an ODE using Inverse Function Theorem

I have the following statement of the inverse function theorem from Munkres' Analysis on Manifolds: (IFT) Let $A$ be open in $\mathbb{R}^n$; Let $f:A\to\mathbb{R}^n $be of class $C^r$. If $Df(x)$ is ...
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Function from $\mathbb{R}^n$ to $\mathbb{R}^n$ with below bounded Jacobian has global inverse

Suppose $f :\mathbb{R}^n \rightarrow \mathbb{R}^n $ is of class $C^1$, and $\|f(x)-f(y)\|\geq\|x-y\|$. Prove that $f$ is global invertible, and $f^{-1}$ is also of class $C^1$. We learnt the ...
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Is this function composed of non-invertible functions invertible?

Consider a nonlinear function $f:\mathbb{R}^3 \rightarrow \mathbb{R}^2$. Next define $g:\mathbb{R}^4 \rightarrow \mathbb{R}^4$ as per: \begin{align} (y_1, y_2, y_3, y_4) = g(x_1, x_2, x_3, x_4) = (f(...
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Does solvability of the differential structure have any implication on solvability of the non-autonomous variant in elementary terms?

To study the differential structure of many functions, like those that come up in the coordinates involving nilpotent matrices, I might implore the inverse function theorem. For example, $f(x) = \frac{...
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On Hessians of inverse vector functions

Let have a function $f:R^{n}\longrightarrow R^{n}$ given by $x_{i}\stackrel{f}{\longrightarrow}y_{i}=y_{i}(x_{1},\ldots,x_{n})$, where $f$ is smooth and non-singular in the domain of interest. The ...
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Necessary condition for Open map in Inverse function theorem

$f:\mathbb{R}^3 \to \mathbb{R}^3$ , $$f(x_1,x_2,x_3)=(e^{x_2cosx_1},e^{x_2sinx_1},2x_1-cosx_3)$$ and $E=(x_1,x_2,x_3)$ such that there exist an open subset U around $$(x_1,x_2,x_3)$$ such that f ...
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Best simulation method in the case of completely specified distribution

Let $f$ be a probability density function of the form $$ f(x)=k \times g(x), $$ where $k$ is the normalizing constant, $g(x)$ is the kernel of the distribution, that is the part which involves $x$, $g(...
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Why is this function the graph of another function?

enter image description here I'm working on the textbook pictured above EDIT: typed out the question here: Let $f: \mathbb{R}^2 \to \mathbb{R}^3$ be a $C^1$ map. Assume $f(0) = (0,0,0)$ and $$\frac{...
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Proof - implicit function theorem $\implies$ inverse function theorem

Let's assume that $f:\mathbb{R}^n\to\mathbb{R}^n$ with \begin{align*} f(y_1,y_2,\cdots ,y_n):=\begin{pmatrix}f_1(y_1,y_2,\cdots ,y_n)\\f_2(y_1,y_2,\cdots ,y_n)\\\vdots\\f_n(y_1,y_2,\cdots ,y_n)\end{...
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Proof of inverse function theorem

I'm reviewing old calculus notes, and we are given the inverse function theorem, note that invertible means injective here, and $f^{-1}:= f^{-1}(f(x))=x, \forall x \in D(f)$. Theorem. If $f$ is ...
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Understanding the Proof of the existence of the Inverse function of a multivariable function.

Introduction: I am having a hard time understanding this Proof. A similar scheme of the proof can be found in countable many books and lectures. However, so far the sources i looked up do not really ...
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Proving every solution to the system of equations lies on a smooth curve

Prove that every point $(x_0,y_0,z_0), 0<y_0<z_0,$ which is a solution to the system $$\begin{cases}x^2+y^2+z^2=2\\x^3+y^3+z^3=1\end{cases}$$ lies on a smooth curve. I interpreted smooth curve ...
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Is this a propper substitution within the inverse function theorem?

Suppose we find an invertible function $f(x)$ and we want to investigate properties of its inverse. The inverse function theorem tells us, if I am interpreting correctly, $\frac{d}{dx}f^{-1}(x) = \...
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local inverse of analytic function

Let $f:R^n\to R^n$ be an analytic map such that $df_a$ is invertible. The "usual" local inversion theorem says that $f$ is locally invertible and that the inverse will be smooth. I'm quite ...
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Why must the inverse function be of this form?

I am reading a proof about the implicit function theorem that uses the inverse function theorem. There they make a statement that I do not understand. The inverse function theorem statement is: ...
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Extensions of the inverse function theorem

Suppose that $g(\cdot, \epsilon) : \mathbb{R}^m\to\mathbb{R}^m$ is invertible. If I have a function $f(x, \epsilon) = g(x, \epsilon) + \epsilon^k e(x, \epsilon)$ for some $k\in\mathbb{N}$ and where $e(...
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Prove Implicit Function Theorem directly from Constant Rank Theorem

For reference: ($\textbf{Constant Rank Theorem}$) Suppose $U_0\subset\mathbb{R}^m$ is open and $F:U_0\rightarrow \mathbb{R}^n$ is a $C^r$ map with constant rank $k$ (that is, its Jacobian matrix has ...
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Asking for clarification on a step for the proof of the inverse function theorem

This is from the book called Calculus and Analysis In Euclidean Space by Shurman et al., pp. 207 Theorem: Let $A$ be an open subset of $\mathbb{R}^n$, and let $f: A \to \mathbb{R}^n$ have continuous ...
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Inverse function theorem on smooth manifolds

Following this thread, I'm trying to prove in detail this theorem. Could you please check if my proof is fine or contains logical mistakes? Theorem: Let $X \subseteq \mathbb R^M$ and $Y \subseteq \...
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If a function has a local inverse everywhere does that mean its invertible?

I just learned the inverse function theorem and I immediately began wondering the following: Let $F:U\to F(U)$ be an $C^1$ function, where $U\subseteq \Bbb R^n$ is an open set. If, for all $x \in U$, ...
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How to get inverse function(series) of a series that with unknown power

I am facing a problem in my research: If a series takes form of: $$ y=a_1x^{n_1}+a_2x^{n_2}+\cdots+a_jx^{n_j}+\cdots $$ How to get its inverse function(series)? Maybe its inverse series takes form of: ...
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Questions about Rudin's rank theorem

I am trying to understand the rank theorem in Rudin's Principles of Mathematical Analysis. The theorem states: Theorem Suppose $m,n,r$ are nonnegative integers, $m\ge r, n\ge r$, $F$ is a $C^1$ ...
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Inverting Jacobian trick: does it always work, or did professor get lucky?

I am tutoring a multivariable calculus student, and his teacher was solving $$\int_R f(x, y) dA$$ using the Change of Variable Theorem, where $f$ is an unimportant function and $R$ is the region ...
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When is a function locally invertible

Let $J\subseteq\mathbb{R}$ be an interval and $f: J\to\mathbb{R}$, $f$ differentiable in $x_0\in J$ with $f'(x_0)\neq 0$. Does there exist a neighborhood of $f(x_0)$ such that $f$ has a continuous ...
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Inverse Function Theorem for Functions That Aren't Continuously Differentiable

I am trying to show that given the vectors $\mathbf{a}$ and $\mathbf{b}$, the system of equations given by $$\mathbf{a}=\mathbf{x}_2 - \mathbf{x}_1$$ $$\mathbf{b}=\frac{\mathbf{x}_1}{\lVert \mathbf{x}...
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Is there a version of the inverse function theorem for a function $f:R^m\rightarrow R^n$ with $n>m$?

in a research project, I need to apply some version of the inverse function theorem to a function $f:R^m\rightarrow R^n$ with $n>m$. More specifically, I have $\gamma=f(t)$, where $\gamma$ has ...
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Second-order envelope theorem for linear programming

Consider parameterized linear programming $V(\theta) = \max_x \langle c(\theta),x\rangle$ s.t. $A(\theta)x\leq b(\theta)$, $x\geq 0$. Let's also assume $c,A,b$ are infinitely differentiable with ...
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Understanding the proof of the Implicit Mapping Theorem

I am following Advanced Calculus of Several Variables by C.H. Edwards, Jr. I failed to build the logic of the theorem III-$3.4$ stated below, Theorem $3.4$: Let the mapping $G: \mathscr{R}^{m+n} \...
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Inverse Function Theorem Exercise

Let $U \subset \mathbb{R^m}$ be an open subset. Let $f:U \to \mathbb{R^n}$ be a $C^1$-smooth map and suppose that $f(x)=0$ for some $x \in U$ and that $Df(x)$ is invertible(derivative map of f at x). ...
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Inverse function consequence in single variable

Suppose we have $f:]a,b[ \rightarrow ]c,d[$ a $C^k$ function such that $f' > 0$. Then $f$ is a bijection and the inverse $f^{-1}$ is $C^k$. ¿How would I prove this as a consequence of the inverse ...
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Inverse function theorem, Tao, Analysis II

In Analysis II by Tao, he wrote: Theorem 6.7.2 (Inverse function theorem). Let $E$ be an open subset of $\mathbf{R}^n$, and let $f : E \to \mathbf{R}^n$ be a function which is continuously ...
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Find the smallest number b so that the function $f(x)=x^3+5x^2+bx+1$ is invertable and evaluate $\frac{d}{dx}f^{-1}(1)$ for that value of b.

I've seen a similar question asked before, but I still can't figure this out. I got $b=\frac{25}{3}$. f(x) has an inverse if f is injective i.e. if $\frac{df}{dx} \geq 0$ for all x. $$3x^2+10x+b \geq ...
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Strong differentiability and the inverse function theorem in Banach spaces

I am trying to prove the strong differentiability version of the Inverse Function Theorem for Banach spaces, but I am not sure if it is true. I am interested in this because it is a kind of punctual ...
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Show that $\gamma$ is a parametrization for the one sheet hyperboloid

I'm given $\gamma:(-\pi,\pi)\times \mathbb{R}\to S$ where $S$ is the one-sheet hyperbolid definied by the equation: $$x^2+y^2-z^2=1$$ Such that $\gamma(u,v)=(\cos(u)-v\sin(u),\sin(u)+v\cos(u),v)$. I'm ...
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$f$ is locally injective, differentiable, is $f$ is injective? [duplicate]

$f$ is locally injective, differentiable, is $f$ is injective? Let $f:U\subset \Bbb R^n\to V=f(U)\subset \Bbb R^n$ be $C^1$, $U=\cup U_i, V_i=f(U_i)$, $f: U_i\to V_i$ is injective, $f'(x)$ is ...
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How to prove there is no continuous differentiable injective map from $\Bbb R^n$ to $\Bbb R$? $n\geq2$. [duplicate]

How to prove there is no continuous differentiable injective map from $\Bbb R^n$ to $\Bbb R$? $n\geq2$. This is a problem in entrance to direct PHD of Tsinghua University. I got it from a webfriend. ...
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confused about the inverse function theorem in Rudin's textbook

The confusion is highlighted with red rectangle.I don't understand the statement "it follows that $\varphi$ has at most one fixed point in $U$ ..." It seems that "The Contraction ...
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The equation $f(\mathbf{x})=\mathbf{c}$ has a solution if $\mathbf{c}\approx\mathbf{0}$ (Munkres "Analysis on Manifolds" Exercise 6 on p.79)

I am reading "Analysis on Manifolds" by James R. Munkres. Is my solution to the following exercise ok? This exercise is in the section 9 "THE IMPLICIT FUNCTION THEOREM". But I did ...
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Does this formula for the derivative of a differentiable inverse have a name?

Consider the following lemma: Let $f:X \to Y$ be an invertible function, with inverse $f^{-1}:Y \to X$,and let $f(x_0) = y_0$. If $f$ is differentiable at $x_0$ and $f^{-1}$ is differentiable at $y_0$...
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Solving $y=x\tan(\theta),f(x,y)=0\implies f(x,x\,\tan(\theta))=0$. Generalized abcissa and ordinate trigonometric functions.

I was inspired to return to an old problem I came up with after seeing this question. This was the problem of finding the analogs of other trigonometric functions which would parametrize a certain ...
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Implications of continuity and invertibility of real functions in reality

Theorem 3.10 in Tom Apostol's Calculus states that for each strictly increasing and continuous function $f$ in the interval $[a, b]$, it's inverse function $g=f^{-1}$ is continuous on $[c, d]$, where $...
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inverse trigonometric functions with plus minus sign

$$ \cos\left(\theta_{} \right) = \pm \frac{ 1 }{ \sqrt{ 3 } } $$ $$ \theta_{} = \pm \cos^{-1} \left( \pm \frac{ 1 }{ \sqrt{ 3 } } \right) $$ I can't get why the leftmost $~ \pm ~...
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Strengthening of Implicit Function Theorem using Second Derivative

The implicit function theorem (let's just say 2-variable functions for now) says that at points at which $\frac{\partial f}{\partial y}$ is non-zero (equiv. the graph doesn't have a vertical tangent ...
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Prove existence of inverse of analytic function in some neighborhood

Suppose $f(z)$ is analytic at $z_{0}$ with $f^{\prime}\left(z_{0}\right) \neq 0 .$ Show that there exists an analytic function $g(z)$ such that $f(g(z))=z$ in some neighborhood of $z_{0}$. This is ...
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Let $f:\mathbb{R}\to \mathbb{R}^2$ be a $C^1$ map. Then the restriction of $f$ to any open set is not surjective

I try to use inverse map theorem in a function defined interns of $f$ or implicit map theorem but I can't get this result.
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Proving a particular case of the complex analytic implicit function theorem using the complex analytic inverse function theorem

In the real analytic case the implicit and inverse function theorem are essentially equivalent, i.e. one can be deduced from the other (from what I know it is more usual to prove the inverse function ...
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apply the Inverse Function Theorem to matrices

Define $f(A) = A^2$ where A is a $n\times n$ matrix. (a) Show that every matrix $B$ in a neighbourhood of the identity matrix $I$ has at least 2 square roots, that is, each varying as a $C^1$ function ...
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