# 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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### 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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### 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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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 ...