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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Lipschitz Continuity of the Value Function [duplicate]

Suppose that $g:[0,1]^2 \rightarrow \mathbb{R}$ is a smooth function. Define the value function $$ g^*(x) = \max_{t \in [0,1]} g(x,t). $$ Question: Under what conditions would $g^*$ be a Lipschitz ...
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38 views

Why is the Lipschitz Condition described using "Cones"?

I have often seen the Lipschitz Condition (Lipschitz Continuity) of mathematical functions being characterized through the following analogy: A function obeys the Lipschitz Condition if each point on ...
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1answer
106 views

Do we have any way of knowing if natural phenomena in the real world follow the "Lipschitz Condition"?

Recently, I keep coming across terms containing "Lipschitz" pertaining to statistical models and machine learning. This includes terms such as "p-lipschitz (rho), lipschitz convexity, ...
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23 views

When is a non-differentiable vector field a gradient field?

I have the following question regarding the connection between vector fields and gradient fields. Assume I have given a vector field $$\textbf{F} : \mathbb{R}^n \to \mathbb{R}^n.$$ I want to know when ...
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1answer
42 views

difference between lipschitz domain and domain satisfying the cone condition

I've come across two definition and I cant understand the differences between them. Maybe someone can help me..Whats the difference between a lipschitz domain and a domain satisfying the cone ...
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50 views

Does the following operation preserve Lipschitz continuity?

For fixed $r>0$, consider $\mathcal{L}: L^\infty(0,1) \rightarrow L^\infty(0,1)$, with $(\mathcal{L}f)(x) = \|f\|_{L^\infty(B_r(x))}$. If $f$ is Lipschitz continuous, is $\mathcal{L}f$ Lipschitz ...
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43 views

If $f$ is periodic and $C^1$ class, then Lipschitz continuous.

I proved if $f : \mathbb R\to \mathbb R$ is periodic and $C^1$ class, then $f$ is Lipschitz continuous. I wonder if my proof is correct and if there is an easier proof. My proof is a little ...
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3answers
51 views

Lipschitz continuity of $f(x)=|x|$ on $[-1,1]$ with period $2.$

Prove the periodic function $f :\mathbb R \to \mathbb R$ s.t. $f(x)=|x|$ on $[-1,1]$ with period $2$ is Lipschitz continuous. Here is my proof. I'm stuck in the last part. Note that $f(x)=|x-2k| \ (...
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12 views

Question on Lipschitz continuity of multiple-input multiple-output function

I can't get an intuitive understanding of Lipschitz continuity for a function with multiple inputs and multiple outputs (MIMO). I am currently working with the Kinky inference (see Callies' article ...
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1answer
20 views

Second Derivative of Lipchitz Concave Curve is infinite at only finite points

Suppose $Q(x):[0,1]\to[0,1]$ is a segment of a convex set which is concave downwards and locally Lipchitz and differentiable a.e. such that $Q(0)=Q(1)=0$ Is $Q''(x)$ going to be tending to $-\infty$ ...
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39 views

A nonnegative, integrable, Lipschitz function $f$ satisfies $\lim \inf_{n \rightarrow \infty} \sqrt{n}f(n) = 0$

Let $f$ be a nonnegative integrable and Lipschitz function in $\mathbb{R}$ with Lipschitz constant $C$. Prove that $\lim \inf_{n \rightarrow \infty} \sqrt{n}f(n) = 0.$ The issue I have with this ...
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31 views

Weak conditons for $(\nabla f(y) - \nabla f(x) )^T (y-x) > 0$

I am looking for conditions on $f$ s.t. $$(\nabla f(y) - \nabla f(x) )^T (y-x) > 0$$ holds true for any $x,y$. Assuming $f$ is twice continuously differentiable and applying the mean-value theorem ...
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31 views

Relationship between smoothness and Lipschitzness

Given a convex function $f:\Omega\mapsto\mathbb{R}$ has function value bounded $|f(x)|\leq B$, diameter of the convex domain bounded $\|x-y\|_2\leq D, x,y\in \Omega$ and $\beta$ smoothness: $\|\nabla ...
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20 views

Lipschitz continuity of $\sqrt{f}$ for $f(x) = \sup_{\alpha \in T} \sum_{i=1}^d \left(\sum_{j=1}^D \alpha_j x_{ij}\right)^2$

Write $x = (x_{ij})_{1\leq i\leq d,1\leq j \leq D} \in \mathbb{R}^{dD}$ and define the function $f ~\colon~ \mathbb{R}^{dD} \to \mathbb R$ by $$f(x) = \sup_{\alpha \in T} \sum_{i=1}^d \left(\sum_{j=1}^...
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44 views

Example of a non-Lipschitz $f \in \mathrm{C}^1(U)$ where $U \subseteq \mathbb{R}^n$ is a non-convex compact connected set

Consider a smooth function of several variables $f: U \to \mathbb{R} \in \mathrm{C}^1(U)$ where $U \subseteq \mathbb{R}^n$ is a connected set. It can be proven using the mean value theorem that if $U$ ...
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1answer
31 views

Proving single solution to an initial value problem

I have the following initial value problem $$ y' = \left|\frac{1}{1+x^2} + \sin{|x^2 +\arctan{y^2}|}\right|, \space y(x_0)=y_0 $$ for each $(x_0, y_0)\in \mathbb{R} \times \mathbb{R}$ I need to prove ...
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40 views

Bounding an $L$-smooth function $f$: $(x-y)^T \left[ \nabla f(x) - \nabla f(z) \right] \geq ? $

Let $f$ be a convex function with $L$-Lipschitz continuous gradient. One can upper bound $( x - y)^T \left[ \nabla f(x) - \nabla f(z)\right]$ by applying Young's inequality followed by Lipschitz ...
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98 views

Given a continous real function, it is almost globally Lipschitz:

Let $f:K\subset\mathbb{R}^N\longrightarrow\mathbb{R}$ continous over $K$ compact. Show that: $$\forall\varepsilon>0\quad \exists L>0\hspace{0.5mm}\mid \hspace{0.5mm}\forall x,y\in K\quad\|f(x)-f(...
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Exterior sphere condition and Lipschitz Condition on the boundary for elliptic PDE

I am currently stuck on a problem. Here is the description: Assume $\Omega$ is a $C^1$, bounded domain that satisfies the exterior sphere condition: For every point $x_0 \in \partial \Omega$, there ...
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38 views

Does this 'cyclic Lipschitzianity' have a name?

I am wondering if the following classes of functions have a name and if they play a role in some branch of analysis. For $L\geq0$, define $C_L$ as $$C_L = \{f:\mathbb R^2\to \mathbb R: |f(x,y)-f(x',y)+...
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32 views

Is the dot product on $\mathbf{R}^n$ ($n\ge 2$) Lipschitz?

Let $n\ge 2$ be some positive integer. Let $f:\mathbf{R}^n\times\mathbf{R}^n\to\mathbf{R}$ be defined as $f(x,y)=x\cdot y$ where $x\cdot y$ denotes the dot product of two vectors. Is $f$ Lipschitz? ...
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Is the multiplication on $\mathbb{R}$ a Lipschitz function?

This question is related to this unanswered question on the site. My question is specifically about the special case when $n=1$: Let $f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ be the function ...
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14 views

Involution of a bilipschitz function is bilipschitz

Set $\phi(x)=\pi^{-\frac{n}{2}}\ e^{-|x|^2}$, $x=(x_1,x_2,\cdots,x_n)$. Assume $f(x)=(f_1,f_2,\cdots ,f_n)$ is a bilipschitz mapping, $F_i(x,t)=(f_i*\phi_{t})(x)$, where $\phi_t(x)=\frac{1}{t^n}\phi(\...
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85 views

Equivalence of norms in the space of Lipschitz continuous functions

I know that norms are not necessarily equivalent in $C[0,1]$ because it is an infinite-dimensional space. For example, I was able to show that $||x||_{\infty}$ and $||x||_2$ are not equivalent in $C[0,...
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27 views

mathematical writing of Lipschitz expressions

Let $f$ be a continuous function $f: \mathbb{R}\times \mathbb{R^n} \rightarrow \mathbb{R^n}$, $(t,y) \rightarrow f(t,y)$ I do not see well how to write mathematically without error the expression: &...
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29 views

Find a Lipschitz constant w.r.t. $y$ of $f(x,y) = \sin(xy)$

Find the Lipschitz constant with respect to $y$ of the function $$ f : [0,3] \times [0,5] \to [-1,1], \qquad (x,y) \mapsto \sin(xy) $$ My solution: $$ \begin{aligned} |f(x,y_1) - f(x,y_2)| &= | \...
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1answer
27 views

Prokhorov or probability metric and (gaussian) convolution as Lipschitz functional

The Prokhorov (or Levy-Prokhorov metric) $d_P$ on the space of probability measures on $\Bbb R$ with respect to the standard metric $d(x,y) = \vert x-y\vert $ is defined as $$d_P (\mu , \nu ) := \inf \...
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1answer
30 views

Riemann sum of Lipschitz function

Let $f : [0,1] →\mathbb{R}$ be a Lipschitz function such that $|f(x) − f(y)| ≤ λ|x − y|$ for all $x,y ∈ [0,1]$. Let ̇$P$ be a tagged partition of $[0,1]$ such that $|| ̇P||< \frac{1}{m}$ for some ...
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1answer
20 views

Proving bounded gradient for composite function

I have a non-convex bounded function $h$ that is L-smooth and it has bounded gradient, i.e. $\|\nabla h(x)\| \leq \sigma^2, \forall x \in \mathbb{R}^d$. Define the function $f(y) = \exp(\alpha \times ...
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1answer
36 views

Use Lipschitz condition to prove some property of the function

Suppose $f(x)$ is differentiable everywhere and satisfy first-order Lipschitz condition, i.e., $$ |f'(x)-f'(y)| \leq \beta \cdot |x-y|, \ \forall x,y $$ Show that $$ f(x)-f(y)-f'(y) \cdot (x-y) \leq \...
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21 views

Function with non-Lipschitzian gradient satisfies descent lemma

It is well-known that if $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is a $C^1$ function with Lipschitz gradient, then $f$ satisfies the descent condition $$f(y)\le f(x)+\left\langle\nabla f(x),y-x\right\...
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1answer
27 views

Continuous mapping theorem for convergence in $L^1$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $K > 0$ such that $|f(x) -f(y)| \leq K \min\{1, |x-y|\}, x, y \in \mathbb{R}$. Also $X_n \xrightarrow[n \rightarrow \infty]{\text{a.s.}} X$. I am ...
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1answer
37 views

Various densities of the set of Lipschitz function

I am curious to know the relationship between the set $Lip(X)$ of lipschitz functions and the all the sets $C(X), C_{0}(X), C^{k}(X), C_{c}(X), C^{\infty}(X), C_{0}^{\infty}(X)$ regarding density. If ...
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1answer
95 views

Strong convexity/Lipschitz gradient duality for convex conjugates and strong convexity/Lipschitz gradient criteria

If $f : \mathbb R^n \to \mathbb R$ is $C^2$ and convex, I want to show that $f$ has a $L$-Lipschitz gradient if and only if its convex conjugate $f^*$ is $\frac{1}{L}$ strongly convex. I received a ...
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36 views

Relation between Lipschitz constant and stability of dynamical system

Consider a nonlinear dynamical system S of the form: $\dot{x}=f(x)$ The function $f$ (and the system S) is said to be Lipschitz if : $\forall x,y \in \mathbb{R}, \| f(x) - f(y) \| \leq K \|x - y \| $ ...
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1answer
86 views

Can second order non-linear ODEs have non-differentiable solutions?

I have been recently exposed to the idea that there can exist solutions to systems with no analytical (symbolic) solutions. However, I could not find any source to the question of whether a second-...
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1answer
61 views

Is gradient of a sigmoid function Lipschitz?

A differentiable function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is said to have Lipchitz continuous gradient or is L-LG if the following holds for some $L>0$: $$ \|\nabla f(\mathbf{x}) - \nabla ...
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1answer
53 views

Show this function is Lipschitz

Consider the following function $$f(x)= \begin{cases} e^{x/2} \qquad x \leq 0 \\ x + e^{-x} \qquad x >0 \end{cases}$$ I need to show it's Lipschitz with constant $1$: $|T(x)-T(y)|<|x-y|$. For ...
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66 views

Approximation of Lipschitz function by smooth functions

I'm reading the book Measure theory and fine properties of functions by Evans and Gariepy, but there is a step in the Sobolev extension theorem (Theorem 4.7) that I don't understand. Concretely, the ...
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35 views

Can i represent derivative as Lipschitz constant, if a function defined on interval (a,b]?

I am working with some piecewise linear functions and its' intervals defined as $I_j$=($x_j$,$x_{j+1}$]. I want to use the derivatives of sub-functions as Lipschitz constants, but in order to do that, ...
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1answer
49 views

Supremum of uniformly Lipschitz functions is Lipschitz

If we have a set of functions $f_i$ such that every $f_i$ has a common Lipschitz constant $M$, is it true that the supremum of these functions at some $x$ is also Lipschitz? Assume that we can ...
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12 views

Locally and globally Lipschitz

I have to see if the function $f(t,x)=x^n, \hspace{2mm}n>1$ and $x\in \mathbb{R}$ satisfies the condition of globally or locally Lipschitz with respect to the second variable $x$. I understand the ...
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21 views

Locally Lipschitz operator in $H^1(0,1) \times H^1(0,1).$

I have been trying to show that a map $T$ defined as $$T(f,g) = \dfrac{f|f|}{g}, \qquad f,g \in H^1((0,1))$$ is locally Lipschitz. I am not very sure how to do it since I am not even sure that $T$ is ...
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75 views

Lebesgue differentiation theorem for monotone functions via Vitali covering lemma

I was reading LECTURES ON LIPSCHITZ ANALYSIS by Juha Heinonen and at the Theorem 3.2 he gives a proof of Lebesgue Differentiation Theorem for monotone functions. He says that we can easily (using the ...
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1answer
23 views

Clarification on proof about Lipschitz function composition

The solution is found here: sum and product of Lipschitz functions I'm not sure from where the following line $$|f(x)g(x) - f(x)g(y)| \leq M|g(x) - g(y)|$$ comes from. I tried looking at assumption ...
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18 views

Question on the distortion of a metric embedding and Lipschitz maps

This is a bit of mild confusion off of Matousek's lecture notes on metric embeddings. An injection between metric spaces $f : X \rightarrow Y$ is a $D$-embedding for some $D \geq 1$ if there is $r >...
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28 views

Lipschitz Constant of the Smooth Minimum

How does one go about finding the Lipschitz constant of the smooth min function (or demonstrating that it is not Lipshitz). My progress is that we define the smooth minimum as follows $$f(\textbf{x}) =...
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17 views

Lipschitz constant dependent on the argument of the function

I am trying to find the strong convex and Lipschitz constants for a function $f: \mathbb{R}^N \to \mathbb{R}$. This involves in upper and lower bounding the eigenvalues of the Hessian matrix $\nabla^2 ...
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29 views

Showing that $C^1(D)\subseteq$ $locallyLipx(D)$ $\subseteq C(D)$

Assume that $(t,x)$ $\in$ $D \subseteq \mathbb{R}\times \mathbb{R}$ $^n$ non-empty, open, and connected set, and that $f:D\longrightarrow \mathbb{R}^n$. Prove that $C^1(D)\subseteq$ $\text{locallyLip}...
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59 views

Lipschitz does not bound increments by 1-variation

Can you find a counterexample to the below claim? Claim: Let $f:\mathbb{R}\to \mathbb{R}$ be Lipschitz, i.e. $$|f(x)-f(y)| \leq K_1 | x - y |$$ for all $x,y$ . Let $x, y :[0,T]\to \mathbb{R}$ be ...

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