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Questions tagged [lipschitz-functions]

For question involving functions satisfying a Lipschitz continuity condition, that is, the distance ratio about the distance of $f(x)$ and $f(y)$ and that of $x$ and $y$ can be bounded independently of $x$ and $y$.

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19 views

How do we show Lipschitz continuity?

I am working on existence and uniqueness of nonlinear systems and one of the prerequisites is the notion of Lipschitz continuity. I am a little bit confused on how we use the definitions to actually ...
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1answer
17 views

Differentiability implies Lipschitz continuity (multivariable)

I am studying from Marsden: Elementary Classical Analysis ($2^{\rm{nd}}$ ed.). I am not able to write down the complete proof of the following theorem (Theorem 6.3.1, page 334). The theorem ...
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1answer
23 views

Is $\sqrt x$ locally Lipschitz continuous everywhere?

Is $f(x)=\sqrt x$ locally Lipschitz continuous everywhere? It is definitely not (globally) Lipschitz continuous. I wonder if it is locally Lipschitz continuous at $x=0$.
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0answers
36 views

Lipschitz continuity of $t|x|^a$

Let $f:\mathbb{R}\times\mathbb{R}\longrightarrow\mathbb{R}$ with $(t,x)\mapsto t|x|^{a}$ be given. Let $a\in[0,\infty)$. Prove that for $a=1$ $f$ is not Lipschitz continuous (wrt 2nd argument) Find $...
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1answer
24 views

Lipschitz constant of a Matrix Valued Function

Consider the function $H(w) = \sum_{i=1}^n f(w^T x_i) x_i x_i^T $, where $w\in \mathbb{R}^d$, $\forall i$: $x_i \in \mathbb{R}^d$, and $f: \mathbb{R} \to [0,1]$. Further, know $|f'(y)| \leq 1$ for all ...
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0answers
17 views

Weird characterization of Lipschitz functions through convex functions

I'm stuck with this problem: Let $L: \mathbb{R} \longrightarrow \mathbb{R}$ be a strictly convex function and assume that the function $u : \mathbb{R} \longrightarrow \mathbb{R}$ satisfies $$L(u(x)) +...
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0answers
12 views

Lipschitz constant for a function of matrix

I have the following function in $\Lambda \in R^{k \times k}$ $$f(X) = (-2X^T\Lambda_1Z^T + 2X^T X\Lambda YY^T)\circ I $$ where $X \in R^{n \times k}$,$Z \in R^{k \times n}$ $Y \in R^{k \times k1}$, ...
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1answer
38 views

Are the bounded Lipschitz functions dense in $L^1(\mu)$?

I am currently reading a paper (L. Ambrosio and B. Kirchheim. Currents in metric spaces) and I stumbled uppon a fact which I don't know how to prove. I have the following setting: Let $X$ be a ...
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0answers
17 views

Directional derivative of a Lipschitz functions exists a.e.

Let $f:\mathbb{R}\to \mathbb{R}$ be any Lipschitz function. We know that it is differentiable almost everywhere (which is basically Radamacher's Theorem in one -dimension). I want to prove the ...
2
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1answer
28 views

How to show locally Lipschitz on $A \subset \mathbb{R}^n$ implies continuity on $A$?

I know this question has been answered before but the way that I am trying to show is different and uses the open balls for the proof. I need a clarification for some part of my proof. Let $A \subset ...
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0answers
22 views

Computing the lipschitz constant of an affine IFS

In Massopoust, Interpolation and Approximation with Splines and Fractals, page 184, the author gives the construction of certain "fractal interpolation functions" as follows: Let $X = [a,b] \times \...
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0answers
14 views

A maximization over Lipschitz functions. Modified from

Here P_r and P_g are two probability distributions. We know that if we ignore the denominator, by Kantorovich's duality, it would be 1-Wasserstein distance between these two distribution. And since ...
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1answer
24 views

Contraction and $\max$ function

$f: \Bbb R \mapsto \Bbb R$ $g: \Bbb R \mapsto \Bbb R$ $h: \Bbb R \mapsto \Bbb R$ $h:=\max\{f(x), g(x)\}$ Is $h$ a contraction on $ \Bbb R$ if $f$ and $g$ are both so? First attempts of ...
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1answer
59 views

Examples of $\alpha$-Lipschitz functions for $ \alpha > 1$.

The question: The function $f(x) = x^\alpha $ belongs to the Lipschitz class of order $\alpha$ gives an example of $\alpha$-lipschitz function for $\alpha \in ]0,1[$. This is $x^{\alpha}$. I guess ...
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0answers
29 views

Derive Lipschitz norm equality.

I am reading the paper "Spectral Normalization for Generative Adversarial Networks". The Lipschitz norm is defined as $$\|f\|_{Lip}=\max \frac{\|f(x)-f(x')\|}{\|x-x'\|}$$ In section 2.1, they claim ...
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1answer
25 views

Lipschitz parameter of tanh function

Problem I am wondering if there is any global Lipschitz parameter for $\tanh(x)=\frac{1-\exp(-2x)}{1+\exp(2x)}$ function. More generally, is there a general way to prove a function is Lipschitz and ...
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29 views

$f: \mathbb R \to \mathbb R, f\in C^1$ then $f$ perserves measurability

There's some ambiguity around these terms so I'll be as clear as I can. A set $A \subseteq R$ is said to be negligible if for all $\epsilon > 0$ there are boxes $Q_1, Q_2, \dots$ such that $A \...
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0answers
49 views

Lipschitz continuity of vector saturation

Let $s: \mathbb{R} \rightarrow [a,b]$ be the saturation function, i.e., $s(x) = a$ if $x \leq a$, $s(x)=x$ if $a < x < b$, $s(x) = b$ if $x \geq b$. Consider the vector saturation function $S: \...
2
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2answers
36 views

Is $f(x)=\cos(\frac {1}{x})$ Lipschitz continuous in $(1, +\infty)$?

I don't know how to prove whether $f(x)$ is or isn't Lipschitz continuous. I tried by using the definition but I can't come up to a conclusion. I'm so bad at this! xD
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2answers
54 views

Does $f(t,y)=4t\sqrt{y}$ satisfy the Lipschitz Condition?

$f(t,y)=4t\sqrt{y}$. Does $f$ satisfy the Lipschitz Condition? In other words how do I check if $$|f(t, y_1) - f(t, y_2)| = |4t\sqrt{y_1} - 4t\sqrt{y_2}| \le C |y_1- y_2|$$ holds. EDIT: My Domain ...
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1answer
21 views

How to show $F(s)=s|s|^{p-2}\log|s|$ is locally Lipschitz?

In a paper I saw this expression: "The function $F:R^*\rightarrow R$ with $F(s)=s|s|^{p-2}\log(|s|), p>1$ is locally Lipschitz." To show this we have to prove that for any $\rho>0$ there ...
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0answers
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Clarification on Some Notation in a Lemma: Approximating LSC functions by Lipschitz continuous functions

I'm reading a book and it states the following lemma: Let $c \in \mathbb{R}, u: E \rightarrow [c, \infty]$ not identically equal to $\infty$ and define for $t > 0$: $$u_t(x) = \inf \{u(y) + t ...
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1answer
9 views

Does Lipschitz continuity of a convex imply boundedness of the domain of its Fenchel conjugate

Let $g:\mathcal{H} \to \mathbb{R}$ be a convex and $L_{g}$-Lipschitz continuous function on a Hilbert space $\mathcal{H}$. Is the domain of its Fenchel conjugate $g^*$, where $$ g^*(y) := \sup_{x \in \...
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1answer
24 views

Density of non-lipschitz functions

Fix a closed interval $I$. Prove that the non-lipschitz functions in $C(I)$ are dense in $C(I)$. I don't even know where to begin! This is a question from a previous exam in a Real Analysis course I ...
2
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0answers
53 views

Lipschitz constant of L2 reg. logistic regression $\sum_i \log \left(1 + \exp\left\{ -t_i \left(w^T x_i\right)\right\} \right) + \mu \|w \|_2^2$

Let the L2 regularized logistic regression function is given by, \begin{align} f(w) &= \frac{1}{N} \sum_i \log \left(1 + \exp\left\{ -t_i \left(w^T x_i\right)\right\} \right) + \mu \|w \|_2^2 \ = \...
1
vote
2answers
48 views

differential equation/Lipschitz

I want to show that the $ (t_- , t_+) $ of $$ y'=\cos y \sqrt{t^2+y^2}, y(0)=\pi$$ is $ (t_- , t_+) =\mathbb R$ Therefore I want to show, that f is Lipschitz. $$ \bigg|\cos y_1 \sqrt{t^2+y_1^2} - \...
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1answer
28 views

Continuity of a function on $\mathbb{R}^n\times\mathbb{R}^n$

Given a function $F=F(x,p):\mathbb{R}^n\times\mathbb{R}^n\to \mathbb{R}$ and provided that for each $p \in \mathbb{R}^n$ $F(\cdot,p)$ is continuous. exist $L>0$ such that $|F(x,p_1)-F(x,p_2)|\leq ...
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1answer
32 views

If $f$ is K-Lipschitz on a closed interval, then $0\le U(f;P_n)-\int_a^bf\le\frac{K}{n}(b-a)^2$

If $f$ is K-Lipschitz on a closed interval $I$ and $P_n$ is a partition in $n$ equal parts, then $0\le U(f;P_n)-\int_a^bf\le\frac{K}{n}(b-a)^2$ Proof: Since $f$ is K-Lipschitz, it satisfies $|f(x)-...
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0answers
24 views

Function which is Lipschitz differentiable

Let us consider $\Phi(\mathbf{x},\mathbf{y})$ which is Lipchitz differentiable. I am wondering if the following expression is whether true or not: $$\Phi(\mathbf{x}_2,\mathbf{y}_2)-\Phi(\mathbf{x}_2,\...
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1answer
28 views

Can the rank of the differential of a Lipschitz map decrease in a small neighbourhood?

Is there an example for a Lipschitz map $f:\mathbb{R}^n\to\mathbb{R}^m$ which is differentiable at $x_o$, with $\operatorname{rank} Df(x_o)=k$, such that there is no open neighbourhood $U$ of $x_0$ ...
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0answers
17 views

Looking for an application of DeMarr's point-fixed theorem

I recently discovered the DeMarr theorem: In a vectorial space, if you got two non expensive maps from a convex compact to itself that commutes, they got a common fixed point. I have no example of any ...
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0answers
25 views

How to demonstrate the uniform continuity of a function in the case it has no limited derivative

First of all, I would like to apologize if I make grammatical errors because English is not my mother tongue. During a lecture at the university my teacher did an exercise that required to find the ...
2
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1answer
26 views

Exercise on Sobolev Spaces: Show that this function belong to $W^{1,\infty}$

Let $u \in C^0(I)$ be a bounded function on the open interval $I=(a,b) \subset \mathbb R$. Suppose that there exists a partition $a=t_0 \lt t_1 \lt \dots \lt t_n=b$ such that: $f \in C^1((...
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0answers
14 views

What is the Maximal Solution of the below Cauchy problem?

Let us define this Cauchy Problem: $$\frac{d x(t)}{dt}=(1+t^2)(\exp(x(t)-1)) \cos(x^2(t)), \quad x(0)=1, (t,x)\in \mathbb{R^2}$$ I have tried to show that ODE have a unique maximal Solution $\phi$ ...
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1answer
19 views

Reference on Lipschitz property of the infimum of a family of Lipschitz functions

I can prove the following fact: the infimum, or supremum, of any family of L-Lipschitz functions is L-Lipschitz, as long as the constant L is fixed. However, since this is a very basic result, I am ...
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0answers
17 views

Globally Lipschitz continuous IVP, approximating a solution with $0<x(0)<1$.

I'm trying to solve this question and I have done it but with $0 \leq x(t) \leq 1$ at the end. Using that f is Lipschitz continuous we know that the solution can't cross and diverge, but it could in ...
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1answer
52 views

Is $\min(f(x),\beta)$ a Lipschitz function?

Let $\beta >0$ be some constant and $f:\mathbb{R}^n \to \mathbb{R}$. Assume that $f(x)$ is $L$-Lipschitz in the domain: $X = \{x \vert f(x) \leq \beta\}$, i.e., $\vert f(x)-f(y) \vert \leq L \vert ...
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1answer
36 views

Existence and uniqueness of solution of $y'=(x-y)^{2/3}$ such that $y(5)=5$

Assume the Initial Value Problem: $$ y'(x)=[x-y(x)]^{\frac23}\equiv f(x,y(x)), \quad y(5)=5 $$ Existence: Since $f: \mathbb{R^2} \longrightarrow \mathbb{R}$ and $(x_0,y_0)=(5,5) \in \mathbb{R^2}$, ...
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0answers
48 views

What is the cheapest known finite dimensional approximation of Lipschitz functions.

Let $Lip_{1}([0,1]^{d})$ the set of all Lipschitz functions on $[0,1]^{d}$ with Lipschitz constant less or equal to $1$. I would like to approximate the set with respect to the uniform topology by a ...
2
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1answer
36 views

Functional Space Inequality for Sobolev Space and Lp Space

Let $X = C_{0}(\Omega) := \{ u \in C(\overline{\Omega})\,|\,u|_{\partial\Omega}=0\}$ and define $F : X \to X$ as Lipschitz continuous function and $F(0) = 0$. Let $\Omega\subset \mathbb{R}^{N}$ be a ...
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2answers
64 views

Bounded Gradient implies Lipschitz proof with the mean value theorem

Let $f:\mathbb{R}^n \to \mathbb{R}$ with $|| \nabla f(x)|| \leq M$ (say it is the Euclidean norm), then f is Lipschitz. I have seen proofs that do this for the case where $f:\mathbb{R} \to \mathbb{R}$...
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3answers
29 views

$f$ is a $k$-Lipschitz continuous function & continuously differentiable $\forall x \in \mathbb{R}^n$. Prove/disprove: $||D_{f}(x)||_{op} \leq k$

EDIT - generalization for vector analysis for $f: \mathbb{R}^n \to \mathbb{R}^m$, $f$ is continuously differentiable $\forall x \ \in \mathbb{R}^n$, and is $k$-Lipschitz continuous function. Prove\...
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1answer
9 views

Non-constant function with 0 Lipschitz semi-norm

Suppose we have a bounded metric space $(X,d)$. We say a function $f:X\to \mathbb{R}$ is Lipschitz if $|f|=\sup_{\substack{x\neq y\\x,y\in X}}\frac{\left|f(x)-f(y)\right|}{d(x,y)}<\infty$. This ...
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0answers
13 views

Is infimum over all Lipschitz constants attained?

Let $E\subset\mathbb R^N$ be non-empty, compact, and convex and let $F : E\to E$ be Lipschitz-continuous with respect to some norm on $\mathbb R^N$. Since all norms on $\mathbb R^N$ are equivalent, $F$...
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0answers
14 views

$V(T(E)) \geq c^n V(E)$

$T:\mathbb R^n \to \mathbb R^n$ is a diffeomorphism such that $\forall x,y \in \mathbb R^n: |f(x)-f(y)| \geq c|x-y|$ $E \subset \mathbb R^n$ is Jordan measurable. Show that $V(T(E)) \geq c^n V(E)$ ...
0
votes
2answers
37 views

lipschitz function in a compact, is it differentiable?

Let $f$ be a lipschitz function in $[0,1]$, (it exists a $C>0$ that we have for all $x,y \in [0,1]$ $|f(x)-f(y)|<C.|x-y|$) Can we prove that $f$ is differentiable ?
0
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1answer
86 views

Lipschitz constant of a matrix

I am studying the Lipschitz continuity and trying to solve the following question: If a function $f(x)= Ax$ is defined for $x \in \mathbb{R}^2$ with $A= \begin{bmatrix} a & b \\ c ...
0
votes
1answer
42 views

Showing function is not Lipschitz continuous

For an analysis exercise, I had to show that the function $\sqrt{1-x^2}$ was uniformly continuous, but not lipschitz continuous on the interval $[-1,1]$. I was able to show it was uniformly continuous,...
0
votes
1answer
26 views

If $f$ has an unbound derivative - is $f$ necessarily non-Lipshitz?

So we assume that $f$ is differentiable on some closed interval $[a,b]$ and has a derivative $f'$ which is not bounded. Is there a way to show that $f$ is non-Lipshitz? If not, is there some ...
2
votes
1answer
39 views

Showing $\{f(\frac{1}{n+1})\}$ converges in $\mathbb{R}$

Question: Let $f:(0,1) \to \mathbb{R}$ be a differentiable function such that $|f'(x)| \leq 5$, for all $x \in (0,1)$. Show that the sequence $\{f(\frac{1}{n+1})\}$ converges in $\mathbb{R}$. ...