1
vote
2answers
43 views

Inverse of 2d function involving sine and cosine

I have the function $f: \mathbb R^2 \to \mathbb R^2$ or more precisely $$f\left([0,\pi/2]^2\right)=\{(x,y) \in \mathbb R^2 : \Vert (x,y) \Vert \leq 1 \text{ and } y\geq0\}$$ which means it is a ...
0
votes
1answer
19 views

Class of the inverse function

The exercise goes like this: Let $f$ be an invertible function of class $C^k([a,b])$, prove that $f^{-1}$ is of the same class. But wait a second: $f(x) = x^3$ is invertible and of class $C^{\infty}$ ...
1
vote
1answer
29 views

Computing the inverse explicitly (real analysis)

I have a function $\ f:\mathbb R\to\mathbb R$ such that $\ f(x,y)=(xe^y,xe^{-y}) $ Let $\ a=(1,0), b=(1,1) $ and let $\ g$ be the continuous inverse of $\ f$ such that $\ g(b)=a$. Compute $\ g$ ...
0
votes
1answer
39 views

Finding the inverse of a function involving |x|

I need to find the inverse of the following function $ f:(-1,1) \rightarrow \mathbb{R} $ $ f(x) = \dfrac{x}{1-|{x}|} $ How do I deal with the absolute value here? Thanks
0
votes
0answers
61 views

Uniform continuity of inverse in only one variable

Let $f:[0,1]\times[0,1]\to \mathbb{R}$ be a (uniformly) continuous functions. Denote the image of $f$ by $D_f:=\{(x,y): x\in[0,1] , 0\leq y \leq f(x,1)\}$ $f$ is such that the section $f_x$, i.e. the ...
1
vote
2answers
43 views

two short doubts about the inverse function in a point

the function is $F(x,y,z)=(y^2+z^2, z^2+x^2, x^2+y^2)$ the point is (-1,1,-1) task: find the local inverse of F in that point. I have already proved that F is actually invertible there. then i ...
0
votes
1answer
40 views

The differentiability class of the inverse function

Here's the final part of a proof (from Marden's Elementary Classical Analysis) of the inverse function theorem, where we have been given that $f$ is of class $C^p$: Could someone please explain the ...
2
votes
0answers
67 views

Continuous function that is invertible in one argument---is its inverse continuous in both arguments?

Suppose that $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a continuous function and that it is invertible in its second argument, i.e. for every $x \in \mathbb{R}$, $f(x,\cdot)$ is invertible with ...
0
votes
0answers
31 views

Find conditions in which $f$ is invertible over its entire image

Let us consider the following bijective applications: $$f_{k}:A_{k}→B_{k}, k=1,2,\ldots,m$$ We construct the following application: $$f=(f_1,f_2,\ldots,f_m):A_1×A_2\times\cdots ...
2
votes
2answers
40 views

How to prove that $f$ is $1-1$ from $E$ on $\{ (s,t) : s> 2\sqrt{t} >0\}$

Question: Let $E=\{(x,y): 0<y<x \}$ set $f(x,y)=(x+y, xy)$ for $(x,y)\in E$ a) How to prove that $f$ is $1-1$ from $E$ on $\{ (s,t) : s> 2\sqrt{t} >0\}$ And how to find formula for ...
6
votes
2answers
488 views

Diffeomorphism from Inverse function theorem

I often heard that it is possible to show by using the inverse function theorem that if a function is smooth(arbitrarily often differentiable, a bijection between open sets and has a non-singular ...
4
votes
2answers
63 views

norm of inverse less than 1

I just wanna ask if what I am doing here make sense: Let $\epsilon$ be arbitrary positive number. Choosing $\epsilon$ and let it approaches 0, I would like to have $||(I-\epsilon A)^{-1}|| < 1$. ...
1
vote
2answers
175 views

Invertible functions in $ R^m$

The definition of an invertible function that my book (Apostol's Mathematical Analysis) gives is: A function $f:S \to\mathbf{R}^n$, where $S$ is open in $\mathbf{R}^n$, has a unique inverse if $f$ is ...
3
votes
0answers
54 views

Equation between the two branches of the lambert w function

Is there an equation connecting the two branches $W_0(y)$ and $W_{-1}(y)$ of the Lambert W function for $y \in (-\tfrac 1e,0)$? For example the two square roots $r_1(y)$ and $r_2(y)$ of the equation ...
4
votes
2answers
149 views

Solve equation $\tfrac 1x (e^x-1) = \alpha$

I have the equation $\tfrac 1x (e^x-1) = \alpha$ for an positive $\alpha \in \mathbb{R}^+$ which I want to solve for $x\in \mathbb R$ (most of all I am interested in the solution $x > 0$ for ...
8
votes
1answer
852 views

Functions whose derivative is the inverse of that function

Everyone knows that there are at least three functions whose derivative is the function itself, namely $e^x, \ 0$ and $-e^{x}$. ( are there more?) I was drawing some polynomials and their ...
3
votes
1answer
164 views

Invertible Derivative

I'm trying to brush up on some differential geometry, but there's a subtle point I don't understand. Suppose $h$ is a diffeomorphism. Then the lecture notes here suggest that it's derivative $df_x$ is ...
3
votes
1answer
181 views

Dense pre-images implies continuous right inverse?

Suppose $f : \mathbb R \to \mathbb R$ is such that pre-image of every point under $f$ is dense in $\mathbb R$. This, of course, implies that $f$ is surjective, and hence has a right inverse ...
1
vote
0answers
158 views

Differentiating an inverse function with respect to a given parameter

Assume we have a real-valued function $f=f_\epsilon(x)$ depending on a (typically small) parameter $\epsilon$, whose inverse (with respect to $x$) is well-defined but hard to compute. What can be said ...