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 $(f^{-1})^{\prime}(f(a))=\cfrac{1}{f^{\prime}(a)}$

273 questions
28
votes
4 answers

A function with a non-zero derivative, with an inverse function that has no derivative.

While studying calculus, I encountered the following statement: "Given a function $f(x)$ with $f'(x_0)\neq 0$, such that $f$ has an inverse in some neighborhood of $x_0$, and such that $f$ is continuous on said neighborhood, then $f^{-1}$ has a…
13
votes
2 answers

Differentiable bijection $f:\mathbb{R} \to \mathbb{R}$ with nonzero derivative whose inverse is not differentiable

I had an exam today, and I was asked about the inverse function theorem, and the exact conditions and statement (as stated in Mathematical Analysis by VA Zorich): Let $X, Y \subset \mathbb{R}$ be open sets and let the functions $f: X \to Y$ and…
11
votes
1 answer

"Counterexample" for the Inverse function theorem

In a lecture we stated the theorem as follows: Let $\Omega\subseteq\mathbb{R}^n$ be an open set and $f:\Omega\to\mathbb{R}^n$ a $\mathscr{C}^1(\Omega)$ function. If $|J_f(a)|\ne0$ for some $a\in\Omega$ then there exists $\delta>0$ such that…
7
votes
0 answers

Prove Implicit Function Theorem directly from Constant Rank Theorem

For reference: ($\textbf{Constant Rank Theorem}$) Suppose $U_0\subset\mathbb{R}^m$ is open and $F:U_0\rightarrow \mathbb{R}^n$ is a $C^r$ map with constant rank $k$ (that is, its Jacobian matrix has constant rank $k$ on $U_0$). Then for any $p\in…
7
votes
2 answers

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
  • 1,601
  • 3
  • 18
  • 42
6
votes
1 answer

Inverse Function Theorem for Manifolds with Boundary as the Domain

In Lee's Smooth Manifolds, it is written that the inverse function theorem can fail for manifolds with boundary. As hint, it is given the inclusion of half space into euclidean space $\iota:\mathbb{H}^n\hookrightarrow\mathbb{R}^n$ My guess is that…
6
votes
1 answer

Inverse Function Theorem for functions $f(x,y)$ and $\int\limits_0^1\frac{\partial f}{\partial x}(tx,y)dt$

I'm struggling with the following problem: Let $f\colon\mathbb{R}^2\to\mathbb{R}$ be a twice continuously differentiable function satisfying $$f(0,y)=0\mbox{ for all }y\in\mathbb{R}$$ (a) Show that $f(x,y) = xg(x,y)$ for all pairs…
Hasek
  • 2,257
  • 11
  • 24
6
votes
2 answers

Spivak, Calculus on Manifolds, Problem 2-37 (b)

2-37 (a) Let $f:\mathbb{R^2}\to\mathbb{R}$ be a continuously differentiable function. Show that $f$ is $not\mbox{ 1-1}$. $Hint:$ If, for example, $D_1f(x,y)\neq0$ for all $(x,y)$ in some open set $A$, then consider $g:A\to\mathbb{R^2}$ defined by…
Ansar
  • 509
  • 3
  • 11
6
votes
1 answer

Is a differentiable (but not $C^1$) function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with invertible derivative everywhere an open map?

Is a differentiable (but not $C^1$) function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with invertible derivative everywhere an open map? I know that if we assume the function is $C^1$, then this is a consequence of the inverse function theorem…
CJD
  • 390
  • 1
  • 11
5
votes
1 answer

Is this result true? Uniform convergence and inverses

Running through some geometry papers, I found some authors use the following idea: Let $f_n : \Bbb C^m \to \Bbb C^m$ be a sequence of holomorphic functions converging uniformly to $f : \Bbb C^m \to \Bbb C^m$ on compact sets. Suppose $f(0) =0$ and…
5
votes
0 answers

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 is a function $g:T\to R$ such that $g(f(x))=x$ and…
5
votes
1 answer

Explanation of "without loss of generality" in an application of Inverse Function Theorem.

Let $U$ be an open subset of $\mathbb R^{n+m}=\mathbb R^n\times \mathbb R^m$ and $g:U\to\mathbb R^m$ a $C^1$ function. Let $p=(x_0,y_0)\in U$ be a point such that $$g'(p):\mathbb R^{n+m}\to \mathbb R^m\text{ is surjective}\tag{$*$}$$ The book I'm…
5
votes
0 answers

Example appliction of Nash-Moser inverse function theorem

I have basic knowledge of PDE and know how to use the standard (Banach space) inverse function theorem to solve $$ -\Delta u+ g(u)=f $$ when $g(0)=g'(0)=0$ and $f$ is small: Define $A(u):=-\Delta u+g\circ u$. Then $A(0)=0$ $A'(0)v=-\Delta v$ is…
5
votes
1 answer

Prove that $f(A)$ is an open set and $f^{-1}:f(A)\to A$ is differentiable.

Let $A\subset \mathbb{R}^n$ an open set, and $f:A\to \mathbb{R}^n$ a one to one and continuously differentiable function so that $\det f'(x)\ne 0$ for all $x\in A$ Prove that $f(A)$ is an open set and $f^{-1}:f(A)\to A$ is differentiable. Any idea…
4
votes
0 answers

Inverse function theorem, Tao, Analysis II

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 differentiable on $E$. Suppose $x_0 \in E$ is such that…
1
2 3
18 19