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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.

For functions of a single variable, the theorem states that if $f$ is a continuously differentiable function with a non-zero derivative at the point $a$, then $f$ is invertible in a neighbourhood of $a$, the inverse is continuously differentiable, and $(f^{-1})^{\prime}(f(a))=\cfrac{1}{f^{\prime}(a)}$

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Inverse Function Theorem and global inverses

We learnt the Inverse Function Theorem for multi-variable functions, and it only dealt with "local" inverses, not "global" inverses. Is my interpretation of a global inverse just that there exists an inverse around ALL points in the domain? Here…
Twenty-six colours
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Step in proof of inverse function theorem

This is a question about a step in the proof of the inverse function theorem. Say we have a function $f\colon V\to W$, that is $C^1$ and where $V,W\subset R^n$. The derivative of $f$ is assumed to be invertible. My book says the following: let…
Sha Vuklia
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What is this notation regarding the Inverse Function Theorem?

A definition in my notes state: If $Df(a)$ is invertible (as a matrix), then $f$ is invertible on an open set $U$ containing $a$. So given that $f(x,y) = (a,b)$ and there exists a $C^1$ local inverse near (x,y) with derivative $Df^{-1}(a,b) =…
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How does inverse function theorem show how interior or boundary points map?

So I've read many references that make proofs about interior points mapping to interior points and boundary points mapping to boundary points and they all seem to cite that the inverse function theorem proves this. However, upon viewing Wikipedia's…
mavavilj
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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, $\frac{d}{dx}f^{-1}(x) = \frac{1}{\frac{d}{dx}f (x)|_{f^{-1}(x)}}$ (weird…
StackQuest
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Show that non-singular is necessary for the Inverse Function Theorem

I am attempting the following problem: Show that the condition that $dF(a)$ be non-singular is necessary in the inverse function theorem by showing that if a function $F$ from a neighborhood of $a$ in $\mathbb{R}^p$ to $\mathbb{R}^p$ is…
Jacob
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Question on the proof of the inverse function theorem

I was reading the proof here for the inverse function theorem. I understand everything except in the middle of page 2, the proof says $$\sum^n_{i=1}-2(f_i(x_0) - y_i) \frac{\partial{f_i}}{\partial{x_j}}(x_0) = 0$$ Since we know that $det(Df(x_0))$…
user1691278
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Part of proof inverse function theorem: show that $D(f^{-1})$ is continuous

This is part of the proof of the inverse function theorem: Let $U,V\subset R^n$ and $f\colon U\to V$ a differentiable bijection. Also assume that $f$ is $C^1$ and $f^{-1}$ is differentiable. Show that $D(f^{-1})$ is continuous. Using the chain…
Sha Vuklia
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Inverse function theorem and finding the open set that allows $f$ to have a local inverse

The inverse function theorem says that there exists an open set $U$ such that there is a local inverse of $f$ around a point $a$. However, how would we actually find an open set containing this point $a$? In particular, the function $$f(x,y) =…
Twenty-six colours
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Showing a function is inversible

Hi I'm struggling with this question. Let $f : ℝ → ℝ$ be defined by $f(x)=\frac{5}{\sqrt {x^3+4}}$. Show that this function is invertable.
Arthur King Lee
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Solve inverse trigonometric equation $\sin\left(\operatorname{cot^{-1}}(x + 1)\right) = \cos\left(\tan^{-1}x\right)$

If $\sin\left(\operatorname{cot^{-1}}(x + 1)\right) = \cos\left(\tan^{-1}x\right)$, then find the value of $x$. Please solve this question by using $\cos\left(\dfrac\pi2 - \theta\right) = \sin\theta$ by changing $\cos\left(\tan^{-1}x\right) =…
Md Sadique
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