# 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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### Wrong proof of Inverse function theorem

Is there anything wrong in my attempt to prove the following section of Inverse function theorem by following: The section I try to prove: Let $f:U\subseteq \mathbb{R}^n\to \mathbb{R}^m$ be ...
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### $f : I \to \mathbb{R}$ differentiable at $x_0 \in I$ but $f^{-1}$ discontinuous at $f(x_0)$ [duplicate]

I need to find a function $f: I \to \mathbb{R}$, where $I$ is an interval (can be open, closed, bounded or unbounded - it doesn't matter), so that $f: I \to f(I)$ is bijective, $f$ is differentiable ...
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### Application of inverse function theorem.

I am working on this problem. Let $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a function given by $f(x) = x + g(x)$, where $g : \mathbb{R}^n \rightarrow \mathbb{R}^n$ is $C^1$. Suppose that for ...
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### Find the inverse of $f(x,y,z)=(x+y+z^2,x-y+z,2x+y-z)$ [duplicate]

This question is linked to Prove that the given function is invertible Find the inverse of $f(x,y,z)=(x+y+z^2,x-y+z,2x+y-z)$. I am looking for an approach to find the inverse of such a 3d function, ...
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### Levenberg-Marquardt algorithm and inverse Jacobian/Hessian

Let’s say I have a function $f(p,q):R^{n+m}→R$, with $p∈R^n$ and $q∈R^m$. I have a set of $q_{i=1,…,k},y_{i=1,…,k}$ and I want to find $p$ so I use the Levenberg-Marquardt algorithm to resolve the ...
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### Prove that if $Df(x)=\eta (x)Dg(x)$, then g is locally a composition of f and some function $\varphi$

I have this problem where I can't figure out the final step: $f,g:U\subseteq\mathbb{R}^n\to\mathbb{R}$ are $\mathscr{C}^1$ functions and $Df(x)\neq 0$ for all $x\in U$. Suppose that there is some ...
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### If $\psi(u,v) = (u, u^2 + v^2, v)$ and $\alpha(t)=(ct - b, 5t^2 - ct + c, t + a)$ is a curve with $\alpha$ in $\text{img}(\psi)$, then $a + b + c$ is

If $\psi(u,v) = (u, u^2 + v^2, v)$ with $(u,v) \in \mathbb{R}^2$ and $\alpha(t) = (ct - b, 5t^2 - ct + c, t + a)$ is a curve with $\alpha \subset \text{img}(\psi)$, then $a + b + c$ is .... Is anyone ...
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### For $F:R^{n} \to R^{m}, m>n$, with local inverse $G$ at $x$ such that $G(F(x)) = x$, is $DG$ a left inverse of $DF$? Why/Why not?

Assuming both functions are differentiable in some neighboorhood, under which conditions, if any, is the Jacobian of $G$ the left inverse of the Jacobian of $F$, ie $(JG)(JF) = Id^{nxn}$? Since we're ...
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### Variant of inverse function theorem

I am trying to work out if the following variant of the inverse function theorem holds: Conjecture: Suppose $f$ is a $C^0$ function, and $f'(0)$ exists and is non-singular. Then there exists a ...
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### Help with proof of Inverse-Function Theorem in $\mathbb R ^ n$

Let $U$ be an open set in $\mathbb{R^n}$, and let $f: U \to \mathbb{R^n}$ be a $C^1$ function such that, for all $x$ $\in$ $U$, $Df(x)$ is an isomorphism. My goal is to show if $f$ is one-to-one, ...
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### Show that $F(t) = (\cos t, \sin t)$ is locally one-to-one but not globally one-to-one.

I am studying Implicit Function Theorem and Inverse Function Theorem. The problem I want to ask is: Show that $F(t) = (\cos t, \sin t)$ is locally one-to-one but not globally one-to-one. I have two ...
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### Inverse implicit function theorem

Let $f:\mathbb{R}^{2}\to \mathbb{R}$ be a twice continuously differentiable function such that $f(0,0)=0$ and $f_{y}(0,0)\ne 0$. By the implicit function theorem, there exists an $\epsilon >0$ and ...
1 vote
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### Continuity of derivative in Inverse Function theorem in Deimling's book

I am having a question on the following proof of the Inverse function Theorem in Deimling's book. It is stated that if $G$ is continuously differentiable, then $G_{|U}^{-1}$ is also continuosly ...
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### Inverse function theorem from Deimling's book

I am looking at the following proof of the Inverse function theorem in "Nonlinear Functional Analysis" by Deimling. Here in the proof, he uses the following argument which seems extremely ...
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