# 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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### Proof that the inverse of an analytic function is analytic which uses only real analysis.

I would like to prove the following result: Let $f:R\to R$ be such that $f(x)=\sum\limits_{k=0}^\infty a_k(x-c)^k$ for all $x$ in some open set $O\subset R$. Suppose that $f'(c)\ne 0$. Then there ...
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### Diffeomorphisms between a nonopen set and an open set of Banach spaces

Let $E_i$ be a $\mathbb R$-Banach space, $U_2\subseteq E_2$ be open and $f$ be a $C^1$-diffeomorphism from $B_1$ onto $U_2$. What do we need to assume in order to conclude that $B_1$ is open? By ...
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### If $\\f:\ U\to\ R^N$ is a submersion, Prove that $g$ is $C^k$

If $\\f:\ U\to\ R^N$ is a submersion of class $C^k$ and $g:f(U)\to\ R^M$ is such that $g\circ f : U\to\ R^M$ is $C^k$ then $g$is $C^k$ In my attempt I know that $D_{f(p_0)}$ is onto, $p_0 \in U$ and ...
Let $\left\{ U_{\alpha },\varphi _{\alpha }\right\}$ be a local chart of a regular surface S $\subset \mathbb{R}^{3}$. That is: $\varphi_{\alpha}:U_{\alpha}\subseteq\mathbb{R}^2\rightarrow \varphi_{\... 0answers 19 views ### convergence to level set Say we have a function$f:\Bbb R^N\to\Bbb R$and a scalar$J$, we define$X := \{x | f(x) = J \}$and assume$X\ne\emptyset$. We have a sequence$\{x_n\}$s. t.$\{J(x_n)\}\to J$. Under what ... 0answers 16 views ### What is solution of$f'(f(x))=\exp(f'^{-1}(x))$with$ f'^{-1}$is a compositional inverse of$f'$? ,Assume$f$a bijective and differentiable function on its domain , I want to find solutions of this functional such that$f\colon\mathbb{R} \to \mathbb{R}$;$f'(f(x))=\exp(f'^{-1}(x))$such that$f'...
Prove that for $(x, y) \in \Bbb R^2$ \begin{cases} x+y+\sin(xy)=2c \\ \sin(x^2 + y) = c^2 \\ \end{cases} can give a solution for all $c ā \Bbb R$ close ...