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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 $y$.

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What is the interpretation / intuition of a lipschtzian function?

What is the interpretation / intuition of a lipschtzian function? I mean, a continuous function is one that "does not show jumps", what is the significance of a function being lipschtziana?
Alexandra Hercilia
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Is the function Lipschitz continuous on $ \overline{B_1(0)} $?

$$ f(x,y)=\frac{2x^2y+y^3}{\sqrt{x^2+y^2}} $$ $$ I \ know \ that \ it \ is \ continuous \ on\ \overline{B_1(0)} \ . \\ I\ think \ I \ have \ to \ show \ ||f(x_1,x_2)-f(y_1,y_2)||\leq L||(x_1,x_2)-(y_1,y_2)|| \ , \ for \ x_i,y_i\in…
Matillo
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Lipschitz constant of a function preserved under equivalent metrics?

Suppose $f$ is a Lipschitz function $(X, d_X) \to (Y, d_Y)$ with Lipschitz constant $K$. Question. If $d_X'$ and $d'_Y$ are metrics that are topologically equivalent to $d_X$ and $d_Y$, is $f$ Lipschitz continuous with the same Lipschitz constant…
Guillaume F.
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Condition for $\phi(x) = x - A^{-1} f(x)$ to be Lipschitz continuous?

Let $A$ be an invertible $p \times p$ matrix, and $\phi(x) = x - A^{-1}f(x) $. Let $M$ be a subset of $\mathbb{R}^p$. Under which condition do we have $\phi(x)$ Lipschitz-continuous with constant $K$ on $M$. That is, for every $(x_1,x_2) \in…
Guillaume F.
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Can a function and its gradient both be globally Lipschitz?

I'm interested in non-trivial examples of a continuously differentiable function $f: \mathbb{R}^d \to \mathbb{R}$ such that both $f$ and its gradient $\nabla f$ are both globally Lipschitz. I could come up with trivial examples where $f$ is…
Krishna Pillutla
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Property in-between Lipschitz continuous and *locally* Lipschitz continuous?

(D1) A function $f$ is called Lipschitz continuous on a set $A$ if there exists $K \ge 0$ such that, for $x, y \in A$, $|f(x_1) - f(x_2)| \le K |x_1 - x_2|$ (D2) A function is called locally Lipschitz continuous on $A$ if for each $x \in A$, there…
Guillaume F.
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Global Lipschitz behavior of given function

Is the function $$ f(x) = sin(x)sgn(x) $$ globally lipschitz? The textbook solutions says so but I've some doubts since $$\frac{\partial f}{\partial x} = 1 \quad\forall x > 0^+ \quad ;\quad = -1 \quad\forall x < 0^-$$ Thus it's not continuous at x…
db18
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How to show the following function is Lipschitz

I am trying to solve the following question but I am constantly obtaining the same incomplete inequality. Let $f:A\subseteq\mathbb{R}^{n}\rightarrow\mathbb{R}$ such that $f(x)=\sup\{\langle x,a\rangle:a \in A\}$. Show that $\left\vert f(x)-f(y)…
bluehills
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How to prove the following function is not Lipschitz continuous?

The function $sin(x^2)$ is not Lipschitz continuous on $\mathbb{R}?$ My steps are like this: Suppose assume for some $C\geq 0$ $$ \bigg|\frac{sin(x^2)-sin(y^2)}{y-x}\bigg|\leq C$$ is true, for all $y\neq x,$ where $x, y\in \mathbb{R}.$ Now how we…
thomus
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Find Lipschitz constant for $F(x,y)=(x(1-ay),y(-1+bx))$

Let $F(x,y)=(x(1-ay),y(-1+bx))$ on $0\leq x,y\leq 10$. Here $a$ and $b$ are assumed to be positive constants. Find the Lipschitz constant on this domain. I have a couple of questions about the solution of this exercise, see below: Q1. How do…
Sha Vuklia
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Non-differentiability implies non-Lipschitz continuity

In a simple one-dimensional framework, it is known that the differentiability of a function (with bounded derivative) on an interval implies its Lipschitz continuity on that interval. However, non-differentiability does not implies non-Lipschitz…
pluton
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Find the Lipschitz constant for the following function

$f(t, \begin{bmatrix}x_1\\x_2\end{bmatrix}, u)=e^{-|u|^2}\begin{bmatrix}x_1\\x_2\end{bmatrix},$ where $f:[0, 3]\times \mathbb{R}^2\times \mathbb{R}\rightarrow \mathbb{R}^2$ is a given nonlinear function. I want to find the Lipschitz constants for…
thomus
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Inverse function theorem with topology

Let's consider an application $T : \mathbb R^n \longrightarrow \mathbb R^n$ such as $T:x \longmapsto x+\phi(x)$ where $\phi: \mathbb R^n \longrightarrow \mathbb R^n$ is $\frac{1}{2}$-Lipschitz continuous. I'm trying to show that T is a bijective…
C.Patrick
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Lipschitz continuity of $f(x,y)=xy^2$

This problem consists of two parts, but I cannot tell the difference between them. Show that $f(x,y)=xy^2$ (a) satisfies a Lipschitz condition on any rectangle $a \le x \le b$ and $c \le y \le d$; (b) does not satisfy a Lipschitz condition on any…
Chad
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How can I prove that $f(x,y)=x^2|y|$ satisfies Lipschitz condition?

on the rectangle $|x| \le 1$, $y \le 1$? My book says that $\frac{f(x,y_1)-f(x,y_2)}{y_1-y_1}$ must be bounded. I got that $\frac{f(x,y_1)-f(x,y_2)}{y_1-y_1}= \frac{x^2(|y_1|-|y_1|)}{y_1-y_2}$ but I don't know how to proceed since if $y_1=y_2$ then…
Chad
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