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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 $y$.

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 $y$.

1807 questions
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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)| &= | \sin(x y_1) -\sin(x y_2)| \\ &= \left| 2 \cos…
Algo
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differential equation/Lipschitz.

Show that $\sqrt{1 + f^2} \in Lip([a, b])$, $\forall f \in Lip([a, b])$. I have an exam tommorrow and i can't wrap my head around this problem. So I was thinking to start with the definiton: $|f(x) - f(y)| \le L|x-y|$ , but doesn't get me anywhere.
antonio
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If $f(x)$ is Lipschitz, is $\langle x, f(x)\rangle$ Lipschitz too?

My question is this: If $x\in \Omega=\{x:x\in\mathbb{R}^n, ||x||=1\} $, and it is given that $f: \Omega\to \mathbb{R}^n$ is Lipschitz, is the quantity $g(x) = x^T f(x)$ (or, for that matter any inner product, $g(x)=\langle x, f(x)\rangle$) Lipschitz…
Nidish Narayanaa
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locally Lipschitz in the second variable

Let $E$ be a normed vector space(or Banach space), $\Omega$ an open set of $\mathbb{R}\times E$ and $f$ a function from $\Omega$ to $E$. We say that $f$ is Lipschitz in the second variable on a subset $W$ of $\Omega$ if there is a constant $k$ such…
ZENG
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Show Lipschitz condition for $f(x) = x+|x|$

I am struggling to prove $$x+|x|$$ to be locally Lipschitz. Since it is not continuously differentiable - not differentiable at $x=0$ - it is hence not globally Lipschitz. But how do I proceed for the local case? $$|f(x) - f(y)| \leq…
mathsnovice
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Lipschitz constant problem

Let $u^{}$ a function in $\operatorname{Lip}_{k}(\Omega)$, that is the set of lipschitz continuous functions from $\Omega \subset \mathbb{R}^n$ to $\mathbb{R}$ with lip constant $\le k$. Denote by $\left[u^{}\right]$ the lipschitz constant of…
user865283
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Compute Lipschitz constant of piecewise linear in $\Bbb R^d$.

Suppose we define the piecewise linear functions in $[a, b]^d$ to be all functions that can be written in the form $f(x) = W_ix + b_i, $ if $ x\in A_i$ where $ \{A_i|i=1, ..., n\}$ is a finite partition of $[a, b]^d$, $W_i\in \mathbb{R}^d$ and…
Lyrios
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Lipschitz function, Berkeley problem 1.2.2

Problem 1.2.2 Suppose that $f$ maps compact interval $I$ into itself and that $$\mid f(x)-f(y) \mid < \mid x-y \mid$$ for all $x, y \in I, x\neq y$. Can one conclude that there is some constant $M<1$ such that, for all $x, y \in I$, $$\mid…
Vinay Deshpande
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If a function is Lipschitz, and differentiable, is its gradient also Lipschitz?

If $f(x)$ is Lipschitz, i.e. $$||f(x) - f(y)|| \le L||x-y||$$ is it's gradient also Lipschitz? $$||\nabla f(x) - \nabla f(y)|| \le K||x - y|| $$ And does $L = K$ ?
Maverick Meerkat
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How to prove this function is not lipschitzianan

I have this exercise for my subject Differentials Equations II: Let $\emptyset\neq I\subset\mathbb{R}$ an interval and $f:\longrightarrow \mathbb{R}$ a function. Let's denote $I^*=\{t\in I:\exists f'(t)\}$ Suppose that exists a sequence $t_n\in…
Antonio Gamiz Delgado
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How can we show that the Log sum exponent function is $L$-Lipschitz continuous?

A Log-Sum-Exponent function of $\mathbf{x}$ is given by: \begin{equation*} \text{lse}(\mathbf{x})=\log\sum_{i=1}^n\exp(x_i) \end{equation*} I have read in literature that it is a contraction but I am not sure how to prove it. My progress so far…
amj
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Lipschitz constant of a function of matrix

The function is given by $f(X) = (AX^{-1}A^\top + B)^{-1}$ where $X$, $A$, and $B$ are $n \times n$ positive definite matrices. I'm trying to find the Lipschitz constant such that $\| f(X)-f(Y) \| \leq L \|X-Y\|$ where $X \geq 0$ and $Y \geq 0$.…
livehhh
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Quotient of Lipschitz functions is lipschitz

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?
Tyler Hills
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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?
David
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Check Lipschitz condition

My mathematical analysis professor gave this exercise to us: check that the function $$f(x)=\frac{x}{1+|x|}$$ verifies the Lipschitz condition globally. Can someone help me to understand how I can reach this result?
Francesco Andreuzzi
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