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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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11 views

Yosida Transform's pointwise convergence.

Let $f = \Gamma-\lim_{j\to\infty} f_{j}(x)$, where $f_{j}:\mathbb{R}\rightarrow [0, +\infty] $ are convex and lower semicontinuous functions such that $\sup_{j} f_{j}(0)< \infty$. I have to prove ...
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9 views

Closedness of Bi-Lipschitz embedding's image

Suppose that $(X,d)$ and $(Y,\rho)$ are complete and separable metric spaces. If $\phi:X\rightarrow Y$ is a bi-Lipschitz embedding, is it the case that $\phi(X)$ is closed in $(Y,\rho)$?
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34 views

Is the metric space of Lipschitz Function with $d_{\infty}$ complete?

This is my real analysis homework. Define $$\mathcal{L} = \{f : [a,b] \to \mathbb{R} : f \text{ is a Lipschitz function }\}$$ and the metric $$d_{\infty}(f,g) = \sup\{|f(x)-g(x)| : x \in [a,b]\}$$ ...
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10 views

Is entropy function lipschitz? [on hold]

Is the information entropy function Lipschitz with respect to $l_{1}$ norm or $l_{2}$ norm?
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1answer
16 views

How to show dual norm of subgradient of a $L$-Lipschitz convex function is bounded by $L$?

I am studying the monograph Online Learning and Online Convex Optimization. At page 133, the author has the following Lemma: $\textbf{Lemma 2.6.}$ Let $f: S \rightarrow \mathbb{R}$ be a convex ...
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2answers
38 views

If $f$ is lipschitz, then $|f(x)| < C \left(1+|x|^λ\right)$

I'm in a first course of analysis and we got this question and I wasn't able to figure it out. Any hint's are welcome. If $f$ is lipschitz, then $\vert f(x)\vert<C(1+\vert x\vert ^λ)$ for some $C,\...
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1answer
20 views

a condition for a tightness of measure

$f: X\to \mathbb R$ be nontrivial continuous, given the fact that $\sup \int\limits_df\mu_n<\infty\forall n$, then could anyone tell me whether $\{\mu_n\}$ is a tight sequence of a probability ...
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17 views

(Lipschitz norm '). Help.

Someone knows how to implement the ∥⋅ ∥ _Lip' (Lipschitz norm ') in Matlab? It's quite difficult, I know that there is a relation between the ∥⋅ ∥ _Lip' and ...
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1answer
37 views

Lipschitz constant for $\|AB - C\|^2_F$ with Lipschitz continuous gradient.

Let there are complex matrices given by $A \in C^{m×n}$, $B \in C^{n \times s}$ and $C \in \mathbb{C}^{m \times s}$ and I have $f(A)= \|AB - C\|_F^2$ which has a gradient given by $g(A) = (AB - C)B^H$...
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9 views

Numerical solution of a locally Lipschitz differential equation. (Gross-Pitaevskii)

A (very simplified) version of the GP-equation, can be written as $\frac{d\alpha}{dt}=-Ui\alpha^*\alpha^2=-Ui|\alpha|^2\alpha$, Where $i$ is the imaginary unit. In contrast to what I first thought, ...
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62 views

Lipschitz constant of $\frac{\|x\|_\infty}{\|x\|_2} x$

Given $x \in \mathbb{R}^n$ with $x \neq \vec{0}$, I want want to find the Lipschitz constant of the map $f(x) = \frac{\|x\|_\infty}{\|x\|_2} x$ In other words I want to find $K$ such that $\| f(x) - ...
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1answer
21 views

What is the intuition behind having upperbound on eigenvalues of Hessian?

Suppose $f: \mathbb{R}^n \rightarrow \mathbb{R}$ and is a $C^1$, i.e., $\nabla f $ is a continuous vector valued function. Show if the $\lambda_{max}$ of $\nabla^2f$ is bounded, then $\nabla f$ is ...
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20 views

Proving Lipschitz Continuity

Let $X$ be a Hilbert Space, $\varphi \in C^{2}(X,\mathbb{R})$, $c\in\mathbb{R}$, and $\varepsilon > 0$. Denote $\varphi^{-1}(\,\cdot\,)$ as the pre-image of $\varphi$ and define \begin{align*} A &...
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25 views

Equivalence of statements regarding convex function with Lipschitz continuous gradient

I came into a practice problem on Lipschitz continuous gradient. Given convex and twice continuously differentiable $f$, prove the following statements are equivalent. $\nabla f$ is ...
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1answer
45 views

Does the boundary of a level set of a Lipschitz continuous function have Lebesgue measure $0$?

Let $f:\mathbb R\to\mathbb R$ be a (bounded, if necessary) Lipschitz continuous function. Are we able to show that $\partial f^{-1}\left(\left\{0\right\}\right)$ has Lebesgue measure $0$. If not, are ...
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How to show $\frac{1}{L}\|f(y)-f(x)\|_2^2 \leq \langle \nabla f(y) - \nabla f(x) ,y-x \rangle$ for strongly convex function?

Suppose $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is a strongly convex function, i.e., $$ \langle \nabla f(y) - \nabla f(x) ,y-x \rangle \geq \theta \|y-x\|_2^2 \,\,\,\,\, \forall x,y \in \mathbb{R}^n ...
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1answer
32 views

Is log-sum-exp a contraction

For $x \in \mathbb{R}^n$, the log-sum-exp (lse) function $\mathbb{R}^n \rightarrow \mathbb{R}$ is defined as $lse(x) = \tau \log \sum_{i=1}^n \exp(\frac{x_i}{\tau})$, where $\tau > 0$ is a ...
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1answer
49 views

Prove a function is Hölder continuous but not Lipschitz continuous

Problem: Given a function $b:\mathbb{R}\rightarrow\mathbb{R}$ defined by $b(x)=3x^{\frac{1}{3}}$. Prove that this function is Hölder continuous but not Lipschitz continuous. My attempt: In order to ...
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1answer
30 views

If $f'$ is Lipschitz continuous with constant $c$, then $f(y)-f(x)-f'(x)(y-x)\ge-\frac c2(y-x)^2$

Let $f\in C^1(\mathbb R)$ and assume $f'$ is Lipschitz continuous with Lipschitz constant $c\ge 0$. How can we show that $$f(y)-f(x)-f'(x)(y-x)\ge-\frac c2(y-x)^2\tag1$$ for all $x,y\in\mathbb R$? I ...
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0answers
6 views

Continuity of map from Lipschitz functions to rough paths

Suppose that $X_t$ is an $\mathbb{R}^d$-valued geometric $p$-rough path, satisfying $$ \Vert \mathbf{X}^j_{s,t} \Vert \leq M(t-s)^{\frac{j}{p}}, $$ for some $M>0$ and every $j \in \{1,\dots,p\}$, ...
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1answer
36 views

Lipschitz function $f^p$

Let $f$ be a Lipschitz function, that is $|f(x)-f(y)|\leq L|x-y|$, and $p>1$. Further suppose that $f$ has compact support. Does it then hold that $f^p$ is also Lipschitz?
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1answer
19 views

Quotient of Lipschitz functions is lipschitz [closed]

I wanna show that the quotient of two Lipschitz functions is Lipschitz. I feel like it should fall out with some kind of basic inequility argument but am struggling with the details. Anyone wanna help?...
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1answer
43 views

Space of All Smooth Lipschitz Functions

This is a followup question to this post. Let $F$ be the collection of all Lipschitz functions from $\mathbb{R}^d$ to itself, which admit $k>0$ derivatives everywhere and such that the $k^{th}$ ...
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1answer
14 views

Lipschitz Functions: Basics

I'm working on a proof in real analysis. Here is the body of the exercise: Let $g: A \rightarrow \mathbb{R}$ be a differentiable function where $g'$ is continuous and $g'(x) < 1$ on a closed ...
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2answers
39 views

Show $f(x)=\frac{x}{1+x^{2}}$ is lipschitz continuous.

I have to show $\frac{x}{1+x^{2}}$ is lipschitz continuous. Hence I have to show $$|f(x)-f(y)|= \left| \frac{x}{1+x^{2}}- \frac{y}{1+y^{2}} \right| < M|x-y|,$$ for some $M \in \mathbb{R}$. I know I ...
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16 views

Lipschitz constant in a bounded open set and infinity sobolev derivative

Let $V$ be a bounded open set. Let $f:V\rightarrow\mathbb{R}$ such that $f\in C(\overline{V})$ and $f$ is Lipschitz function on V ($Lip(f,V)<+\infty$). I would like to show that $Lip(f,\overline{V})...
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1answer
23 views

Why $f(x)=x^2$ is local Lipschitz, general question about local/global Lipschitz.

I'm trying to understand the difference between global lipschitz and local lipschitz. Let $f(x)=x^2$ while $x \in \mathbb{R}$ if we look at global lipschitz, for all $M \subset \mathbb{R} \times \...
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10 views

Lipschitz Mapping of $\mathbb{R}_+$ to $[0,1)$

During my research, I encountered a problem where I have some points in $\mathbb{R}^2_+$ (actually, there are in $\mathbb{N}^2$), and I would like to map them into the two-dimensional square $[0,1)^2$ ...
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1answer
27 views

Continuity and Lipschitz Property of Infimal Convolution

If $f$ is convex and Lipschitz-continuous on a real Hilbert space $H$ and $g$ is lsc and convex then is the infimal convolution $f\square g$ Lipschitz?
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1answer
33 views

How to prove that a function $f\colon c_0\to c_0$ is not Lipschitz continuous?

I wonder if the following function $f\colon c_0\to c_0$ ( $c_0$ is a space of real sequences convergent to 0 with supremum norm) $$ f(x)=(f_{n}(x)),$$ where $f_n(x)=\sqrt{|x_n|}+\frac{1}{n+1}$ is ...
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1answer
36 views

Proving a Lipschitz inequality for continuously differentiable functions.

So I'm trying to show that given $\phi:G\to\mathbb{R}^n$ where $G\subseteq\mathbb{R}^n$ continuously differentiable, that if $\sup_G\|D\phi\|<c$, then $\phi$ is Lipschitz with constant $c.$ I'm a ...
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1answer
30 views

Lipschitz Constant of a linear function

What is the Lipschitz constant of a linear function, in the form of f(x)=ax+b For any p,q in the domain, ||f(p)-f(q)|| = ||(ap+b) - (aq+b)|| = ||a(p-q)|| <= |a|*||p-q|| Is it a?
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2answers
32 views

Correct way to show that a Lipschitz condition is satisfied

In one of my homework problems, I have an IVP $$ y'=e^{t-y}, \hspace{5mm}where\hspace{3mm}0≤t≤1,\hspace{3mm}y(0)=1 $$ And I need to show that $$ f(t,y)=e^{t-y} $$ satisfies a Lipshitz condition. To ...
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2answers
23 views

pointwise approximation to the identity

Let $\mu$ be a positive Borel measure on $\mathbb{R}$ with $\int_{\mathbb{R}}\mathrm{d}\mu=1$. For $\varepsilon>0$ define the measures $\mu_\varepsilon$ by $$\int f\mathrm{d}\mu_\varepsilon=\int f(\...
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1answer
20 views

Lipschitz condition on $x*|y|$

The IVP $y'=x|y|$ is given along with the condition $y(1)=0$. Upon checking the Lipschitz condition one gets, $|x|*||y_{2}|-|y_{1}||\le|x|*|y_{2}-y_{1}|$ Now, if I go locally around $x=1$, $|x|$ ...
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0answers
28 views

Is a composition of a Lipschitz continuous function with a Continuously Differentiable function Lipschitz over an unbounded set?

Assume the following: $U \subseteq \mathbb{R}^m$ is a closed and connected set (not necessarily bounded), $\Xi \in Lip_{\alpha}(U,X)$, where $X \subseteq \mathbb{R}^n$, $g \in C^1(X,Y)$, where $X \...
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1answer
73 views

An example of a function in weighted Lipschitz class but not in Lipschitz class

For $p \ge 1,$ let $L^p[0,2\pi]$ be the space of $2\pi$-periodic measurable real valued functions. The norm in $L^p[0,2\pi]$ is defined as $$\|f\|_p = \left(\frac{1}{2\pi}\int_0^{2\pi}|f(x)|^pdx\...
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24 views

Best Power in a Probability Inequality

Let $f:S^{n-1}\rightarrow \mathbb{R}_{+}$ be a Lipschitz function. For $1\leq k\leq n$, define $f_k:G_{n,k}\rightarrow \mathbb{R}_{+}$ by $f(E)=\max_{x\in S^{n-1}\cap E}f(x)$. Let $\sigma_k$ denote ...
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35 views

Why inverse Mills ratio for normal distribution is 1-Lipschitz continuous?

The inverse Mill ratio for a standard normal distribution is: $$ IMR(x) = \frac{\phi(x)}{\Phi(x)}, $$ where $\phi(x)$ is the pdf of standard normal distribution and $\Phi(x)$ is the cdf of standard ...
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1answer
26 views

Prove that a Lipschitz function is continuous. Must a Lipschitz function necessarily be differentiable?

I have: Pf. Let epsilon>0 be given. Select a delta such that delta is less than epsilon/M so that |x-y| is less than delta. Then|(f(x)-f(y)| is less than M|x-y| is less than M(delta) is less than ...
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1answer
63 views

Bound on hessian when Lipschitz gradient is bounded

I know that if we have a Lipschitz gradient $$\|\nabla f(x) - \nabla f(y)\|\leq L\|x-y\|,\, \forall x,y, $$ we can say that $\nabla^2f\preceq LI.$ I have a problem where difference of gradient is ...
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96 views

Lipschitz constant of the exponential map

Let $M$ be a smooth Riemannian manifold and let $p \in M$. Suppose $r \ll \text{inj}(p)$ (the injectivity radius at $p$) and fix $t \in (0,r)$ then define the map $$ T_pM \ni v \mapsto \exp_p(tv) \in ...
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0answers
74 views

Lipschitz continuous and Jacobian matrix

Consider a function $f:\mathbb{R}^n\longrightarrow\mathbb{R}^m$ with partial derivatives everywhere so that the Jacobian matrix is well-defined. Let $L>0$ be a real number. Is it true that: $$|f(x)-...
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2answers
35 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
52 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
37 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
42 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
30 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
39 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
34 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 $X \in R^{n \times k}$,$Z \in R^{k \times n}$ $Y \in R^{k \times k1}$, ...