# 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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### Is it possible to find the inverse of the following function explicitly?

I am interested in the following function $$f(x) = \sqrt{3x \left(1-\frac{x}{\theta} \right) \ln{\left[a \left(1-\frac{\theta}{x} \right) \right]}},$$ with $0<a \leq 1$ and $\theta<x$. ...
1 vote
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### Manifolds - Inverse Function Theorem Form?

For context, I am reading Appendix F is Milnor's book on Sullivan's No Wandering Theorem and this is where he uses the following manifolds result: Let $F:X \rightarrow Y$ be a smooth function between ...
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### Proof verification: $dF_p$ nonsingular, then $F(p)\in \textrm{Int}N$

I am trying to solve Problem 4-2 on Lee's Introduction to Smooth Manifold. The problem states the following: Suppose $M$ is a smooth manifold (without boundary), $N$ is a smooth manifold with ...
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### Implicit function theorem by only one partial derivative is continuous.

Question: Let $f \colon \mathbb{R}^2 \to \mathbb{R}$ be a continuous function so that $\frac{\partial f}{\partial y}$ exists and is continuous on $\mathbb{R}^2$. Let $(x_0, y_0) \in \mathbb{R}^2$ be ...
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### Proving a set is a regular surface using inverse function theorem

I was recently introduced to a particular definition of a regular surface which is: A (regular) surface is a subset $S \subset \mathbb{R}^{3}$ with the property that around every point $p \in S$ we ...
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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 ...
1 vote
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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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### Prove that $\varphi\circ X_1$ is also a parametrization if $\varphi$ is a diffeomorphism and $X_1$ is a parametrization

Suppose that $X_1:U_1\subset R^2 \to S_1$ and $\varphi:S_1 \to S_2$ is a diffeomorphism, I want to prove that $\varphi \circ X_1:U_1 \to S_2$ is a parametrization of $S_2$ Here is my attempt: Suppose ...
1 vote
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### Analytical computation of an inverse function

A function defined as $f(x,y)=0$ is said to be implicitly defined by the equation. The concept of inverting $f$ is closely related to the idea of solving the equation $y=f(x)$ for x as a function of y....
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### If $f$ is continuously differentiable, $f'(x)$ is invertible, and $V$ is open, then $f(V)$ is open

This is an exercise from Tao's analysis chapter $6$. Exercise $6.7.3$. Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a continuously differentiable function such that $f'(x)$ is an invertible linear ...
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### Spivak's proof of the Inverse Function Theorem

I recently read Spivak's proof of the Inverse Function Theorem (in his Calculus on Manifolds), and now I've been trying to repeat it without the assumption that the total derivative of $f$ at $a$ is ...
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### Non-singular Jacobian at a point means there exists a region where Jacobian is non-singular

I am currently reading through Wade's introduction to analysis and am having some confusion with a lemma before the inverse function theorem. We have by hypothesis that f is $C^1$ on an open set $V$ ...
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According to the inverse function theorem, a continuous, differentiable function from $\mathbb{R} ^n \mapsto \mathbb{R}^n$ is invertible if the determinant of the mapping function's Jacobian matrix is ...