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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Inverse function theorem on smooth manifolds

Following this thread, I'm trying to prove in detail this theorem. Could you please check if my proof is fine or contains logical mistakes? Theorem: Let $X \subseteq \mathbb R^M$ and $Y \subseteq \...
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If a function has a local inverse everywhere does that mean its invertible?

I just learned the inverse function theorem and I immediately began wondering the following: Let $F:U\to F(U)$ be an $C^1$ function, where $U\subseteq \Bbb R^n$ is an open set. If, for all $x \in U$, ...
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How to get inverse function(series) of a series that with unknown power

I am facing a problem in my research: If a series takes form of: $$ y=a_1x^{n_1}+a_2x^{n_2}+\cdots+a_jx^{n_j}+\cdots $$ How to get its inverse function(series)? Maybe its inverse series takes form of: ...
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Questions about Rudin's rank theorem

I am trying to understand the rank theorem in Rudin's Principles of Mathematical Analysis. The theorem states: Theorem Suppose $m,n,r$ are nonnegative integers, $m\ge r, n\ge r$, $F$ is a $C^1$ ...
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Inverting Jacobian trick: does it always work, or did professor get lucky?

I am tutoring a multivariable calculus student, and his teacher was solving $$\int_R f(x, y) dA$$ using the Change of Variable Theorem, where $f$ is an unimportant function and $R$ is the region ...
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When is a function locally invertible

Let $J\subseteq\mathbb{R}$ be an interval and $f: J\to\mathbb{R}$, $f$ differentiable in $x_0\in J$ with $f'(x_0)\neq 0$. Does there exist a neighborhood of $f(x_0)$ such that $f$ has a continuous ...
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Inverse Function Theorem for Functions That Aren't Continuously Differentiable

I am trying to show that given the vectors $\mathbf{a}$ and $\mathbf{b}$, the system of equations given by $$\mathbf{a}=\mathbf{x}_2 - \mathbf{x}_1$$ $$\mathbf{b}=\frac{\mathbf{x}_1}{\lVert \mathbf{x}...
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Is there a version of the inverse function theorem for a function $f:R^m\rightarrow R^n$ with $n>m$?

in a research project, I need to apply some version of the inverse function theorem to a function $f:R^m\rightarrow R^n$ with $n>m$. More specifically, I have $\gamma=f(t)$, where $\gamma$ has ...
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Second-order envelope theorem for linear programming

Consider parameterized linear programming $V(\theta) = \max_x \langle c(\theta),x\rangle$ s.t. $A(\theta)x\leq b(\theta)$, $x\geq 0$. Let's also assume $c,A,b$ are infinitely differentiable with ...
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Understanding the proof of the Implicit Mapping Theorem

I am following Advanced Calculus of Several Variables by C.H. Edwards, Jr. I failed to build the logic of the theorem III-$3.4$ stated below, Theorem $3.4$: Let the mapping $G: \mathscr{R}^{m+n} \...
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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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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 ...
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Find the smallest number b so that the function $f(x)=x^3+5x^2+bx+1$ is invertable and evaluate $\frac{d}{dx}f^{-1}(1)$ for that value of b.

I've seen a similar question asked before, but I still can't figure this out. I got $b=\frac{25}{3}$. f(x) has an inverse if f is injective i.e. if $\frac{df}{dx} \geq 0$ for all x. $$3x^2+10x+b \geq ...
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Strong differentiability and the inverse function theorem in Banach spaces

I am trying to prove the strong differentiability version of the Inverse Function Theorem for Banach spaces, but I am not sure if it is true. I am interested in this because it is a kind of punctual ...
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Show that $\gamma$ is a parametrization for the one sheet hyperboloid

I'm given $\gamma:(-\pi,\pi)\times \mathbb{R}\to S$ where $S$ is the one-sheet hyperbolid definied by the equation: $$x^2+y^2-z^2=1$$ Such that $\gamma(u,v)=(\cos(u)-v\sin(u),\sin(u)+v\cos(u),v)$. I'm ...
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$f$ is locally injective, differentiable, is $f$ is injective? [duplicate]

$f$ is locally injective, differentiable, is $f$ is injective? Let $f:U\subset \Bbb R^n\to V=f(U)\subset \Bbb R^n$ be $C^1$, $U=\cup U_i, V_i=f(U_i)$, $f: U_i\to V_i$ is injective, $f'(x)$ is ...
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How to prove there is no continuous differentiable injective map from $\Bbb R^n$ to $\Bbb R$? $n\geq2$. [duplicate]

How to prove there is no continuous differentiable injective map from $\Bbb R^n$ to $\Bbb R$? $n\geq2$. This is a problem in entrance to direct PHD of Tsinghua University. I got it from a webfriend. ...
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confused about the inverse function theorem in Rudin's textbook

The confusion is highlighted with red rectangle.I don't understand the statement "it follows that $\varphi$ has at most one fixed point in $U$ ..." It seems that "The Contraction ...
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The equation $f(\mathbf{x})=\mathbf{c}$ has a solution if $\mathbf{c}\approx\mathbf{0}$ (Munkres "Analysis on Manifolds" Exercise 6 on p.79)

I am reading "Analysis on Manifolds" by James R. Munkres. Is my solution to the following exercise ok? This exercise is in the section 9 "THE IMPLICIT FUNCTION THEOREM". But I did ...
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Does this formula for the derivative of a differentiable inverse have a name?

Consider the following lemma: Let $f:X \to Y$ be an invertible function, with inverse $f^{-1}:Y \to X$,and let $f(x_0) = y_0$. If $f$ is differentiable at $x_0$ and $f^{-1}$ is differentiable at $y_0$...
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Solving $y=x\tan(\theta),f(x,y)=0\implies f(x,x\,\tan(\theta))=0$. Generalized abcissa and ordinate trigonometric functions.

I was inspired to return to an old problem I came up with after seeing this question. This was the problem of finding the analogs of other trigonometric functions which would parametrize a certain ...
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Implications of continuity and invertibility of real functions in reality

Theorem 3.10 in Tom Apostol's Calculus states that for each strictly increasing and continuous function $f$ in the interval $[a, b]$, it's inverse function $g=f^{-1}$ is continuous on $[c, d]$, where $...
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inverse trigonometric functions with plus minus sign

$$ \cos\left(\theta_{} \right) = \pm \frac{ 1 }{ \sqrt{ 3 } } $$ $$ \theta_{} = \pm \cos^{-1} \left( \pm \frac{ 1 }{ \sqrt{ 3 } } \right) $$ I can't get why the leftmost $~ \pm ~...
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Strengthening of Implicit Function Theorem using Second Derivative

The implicit function theorem (let's just say 2-variable functions for now) says that at points at which $\frac{\partial f}{\partial y}$ is non-zero (equiv. the graph doesn't have a vertical tangent ...
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Prove existence of inverse of analytic function in some neighborhood

Suppose $f(z)$ is analytic at $z_{0}$ with $f^{\prime}\left(z_{0}\right) \neq 0 .$ Show that there exists an analytic function $g(z)$ such that $f(g(z))=z$ in some neighborhood of $z_{0}$. This is ...
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Let $f:\mathbb{R}\to \mathbb{R}^2$ be a $C^1$ map. Then the restriction of $f$ to any open set is not surjective

I try to use inverse map theorem in a function defined interns of $f$ or implicit map theorem but I can't get this result.
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Proving a particular case of the complex analytic implicit function theorem using the complex analytic inverse function theorem

In the real analytic case the implicit and inverse function theorem are essentially equivalent, i.e. one can be deduced from the other (from what I know it is more usual to prove the inverse function ...
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apply the Inverse Function Theorem to matrices

Define $f(A) = A^2$ where A is a $n\times n$ matrix. (a) Show that every matrix $B$ in a neighbourhood of the identity matrix $I$ has at least 2 square roots, that is, each varying as a $C^1$ function ...
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A Question about Graph of Function under Diffeomorphism

I have a silly question about the graph of a function, which I accidentally encountered in physics. Consider the graph of the function $q(t)$ in the $q-t$-plane, then it can also be viewed as a curve $...
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Reference to the Inverse Mapping Theorem involving Banach spaces

I would like a reference that has the proof of the following theorem: Inverse Mapping Theorem: Let $X,Y$ be two Banach spaces, $\Omega \subseteq X$ an open subset and $a\in \Omega$. Suppose that $f:\...
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Continuous $f:$ open $\Omega \mapsto Y, Y$ (Banach) $\Rightarrow$ Inverse function is an open map

I'm self-studying Cheney's book in Functional Analysis, and this is among the exercises (problem 3.4.5). I think I've proven the claim in the problem, but I guess I've made errors along the way or ...
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Find derivative using Inverse Function Theorem

If $f(u,v)=(3u^2-v,2u^4+v^3)$ then show that $f^{-1}$ exists and is differentiable in some nbd. of $(1,2)$ and find out $D(f^{-1})(1,2)$. Clearly $f$ is contnuously differentiable and we get, $$Df(1,...
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Finding Local Inverse of Transcendental Function

Disclaimer: This is a homework problem for my graduate Applied Complex Variables course. The book being used is "Applied and Computational Complex Analysis: Volume 1" by Peter Henrici. ...
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Application of the inverse function theorem on a convex set

Let $f:U \rightarrow\mathbb{R}^{m}$ continue in a convex set $U \subset \mathbb{R}^{m}$. If $\langle{f'(x)(v),v}\rangle>0$ for all $x \in U$ and for all $0 \neq v \in \mathbb{R}^{m}$, then $f$ is a ...
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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 \...
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Proof of continuity of a map from product of two spaces to $\mathbb R^n.$

$\mathbf {The \ Problem \ is}:$ Given , $U$ is an open subset of $\mathbb R^{m+n}$ and $f : U \to \mathbb R^n$ be a $C^1$ map ,then show that the map $F : (p,v) (\in U ×\mathbb R^{m+n}) \mapsto Df_p (...
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Compute $Df^{-1}(2,5)$ given $f(u,v)=(uv,u^2+v^2)$ using the Inverse Function Theorem (IFT)

I need help figuring out where my solution below goes wrong, because the question was phrased as if the derivative would exist. $$f(u,v)=(uv,u^2+v^2)$$ We prove $D {f}^{-1}(2,5)$ does not exist. ...
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How can I get an intuition about Copula?

I am really struggling with getting an intuition about copulas. I have red many articles and I am stuck at what is the concept/idea behind it. For example if I have two random variables X and Y and I ...
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Reference request: Variations of inverse function theorem

I have two real and separable Hilbert spaces $X$ and $Y$ as well as a continuously Fréchet differentiable function $f : X \to Y$ that is bijective. Furthermore, at every point $x \in X$, $D_x f \in \...
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Global invertibility of a given function

Say we have $f:\Bbb{R}\rightarrow \Bbb{R}$ s.t $f\in C^\infty$ and for every $x\in \Bbb{R}$, $|f'(x)|<1$. I have to prove that the function $F(x,y)=(x+f(y), y+f(x))$ is locally invertible for every ...
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connected components of domain without preimage of critical values

Let $f:M \to N$ be a differentiable map between two connected compact manifolds both of dimension $n\ge 2$ (I think i know the answer to my question if $n=1$). Let us assume that both $M$ and $N$ are ...
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If $\varphi(x,y)=(x+f(y),y+f(x))$ with $|f'(t)|\le k<1$ for all $t\in\mathbb{R}$, then $\varphi$ is a diffeomorphism

Let $f:\mathbb{R}\to\mathbb{R}$ a $C^1$ function, such that $|f'(t)|\le k<1$ for all $t\in\mathbb{R}$. Define $\varphi:\mathbb{R}^2\to\mathbb{R}^2$ by, $\varphi(x,y)=(x+f(y),y+f(x))$. Show that $\...
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Explaining an incorrect solution involving derivative of an inverse.

Suppose $f$ is injective and $f(1)=5$ and $f'(1)=3$. Then a quick application of the inverse function theorem gives $(f^{-1})'(5)=1/3$. However, what is incorrect in the following?: $$(f^{-1})'(5)=(f^{...
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inverse image of a map on a matrix

I have the following mapping and concrete matrices $A \in \mathbb{Q}^{3,4}$ and $B \in \mathbb{Q}^{3,3}$ $$f:\mathbb{Q}^{4,3} \rightarrow \mathbb{Q}^{3,3}, X \mapsto AX$$ I have to find $f^{-1}(B)$ ...
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Surface in $\mathbb{R}^3$ implicitely defined by a $C^1$-function is locally similar to $xy$-plane

Let $F:\mathbb{R}^3\to \mathbb{R}$ be a $C^1$-function and suppose that $(dF)(x,y,z)\not=0$ wherever $F(x,y,z)=0$. Call $$O = \{(x,y,z)\mid F(x,y,z)=0\}.$$ Then, for every point $p\in O$, there is an ...
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87 views

Prove that there is an $n \times n$ matrix $B$ such that $A=B^{2}+B .$

Given a positive integer $n,$ prove that there is $\varepsilon>0$ such that for every $n \times n$ matrix $A$ with $|A|<\varepsilon$ (Hilbert-Schmidt norm), there is an $n \times n$ matrix $B$ ...
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Assumption in prooving the Inverse Function Theorem (in Spivak's "Calculus on manifolds")

My question follows up with an additional remark from Spivak's proof of Inverse Function Theorem. The problem I have is the statement which immediately follows the If the theorem is true for $λ^{−...
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For any point in $\Bbb R^2$ $f$ has a local inverse. Can I say $f$ is invertible on its range?

Define $f:\Bbb R^2\to \Bbb R^2$ by $$\begin{align}f(x,y)=(e^{2x+y},4x^2+4xy+y^2+6x+4y)\ . \end{align}$$ Define $U:=f(\Bbb R^2)$. Prove that the inverse function of $f$ (say $f^{-1}:U\to \Bbb R^2$) ...
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How do I find the Inverse of $f(x)$?

Given $f(x+1)+f(x-1)=x^2$ I have subtituted $(a=x+1)$ and $(a=x-1)$ and got $$f(x)+f(x-2)=(x-1)^2 \text{ and } f(x+2)+f(x)=(x+1)^2$$ Combining those equations, I got $$f(x+2)-f(x-2)=4x$$ I could not ...

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