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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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How to prove that $\sin x$ is a lipschitz continuous function on the real line?

How to prove that the function $\sin x$ is a lipschitz continuous function on the real line using the definition of lipschitz continuity. I know $|\frac{df}{dx}|\leq 1.$ But I wanted to prove without using this condition.
thomus
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Is this function satisfy the Lipschitz condition?

Does the function $F(x,y) = xy^1/3$ satisfy the Lipschitz condition on the rectangle $ {(x,y) : |x| \le h, |y| \le k} $ where $h < 0$ and $k < 0$? I have tried using the mean value theorem to show this: |F(x, u) − F(x, v)| = |$F_y (x, w)$ (u − v)| ≤…
mathsy123
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Canonical Predual of Lipschitz spaces

Suppose $M$ is a subset of a Banach space $X$. Here, page $179$, the author defined Lipschitz-free space $\mathcal{F}(M)$ to be the canonical predual of the space of Lipschitz functions Lip$(M)$, i.e. the closed linear span of the point…
Idonknow
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Lipschitz constant and Rademacher's theorem

When reading a paper, I found a theorem stated as Theorem 1 (Radmacher, [22, Theorem 3.1.6]) If $f:\mathbb R^n\to\mathbb R^m$ is locally Lipschitz continuous function, then $f$ is differentiable a.e. Moreover, if $f$ is Lipschitz continuous,…
govindah
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Proof of Rademacher’s theorem

I read two proofs of Rademacher’s theorem, on the book Measure Theory and Fine Properties of Functions by Evans, P103 and Sets of Finite Perimeter and Geometric Variation Problems by Francesco, P75. (1) On Evans’s book, there is a step to derive…
MichaelS
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How to find lipschitz constant of the gradiant of $\sqrt{1+x^2}$?

I know $\left|\frac{x}{\sqrt{1+x^2}}\right| \le 1$. But I don't know how to find $L$ that $\left|\frac{x}{\sqrt{1+x^2}}-\frac{y}{\sqrt{1+y^2}}\right| \le L|x-y|$. Would you please explain it? Thank you.
Mina
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Multivariable Lipschitz continuous functions that can be written as a linear combination of a function

Let $f_i$ ($i=1, \ldots, n$) be globally Lipschitz continuous functions which can be expressed as $\left[\begin{array}{c} f_1(x_1, \ldots, x_n) \\ \vdots \\ f_n(x_1, \ldots, x_n)\end{array}\right] = \left[\begin{array}{ccc} a_{11} & \cdots & a_{1n}…
Jason
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Lipschitz constant of rescaled function

Let $f:\mathbb{R}^n \to \mathbb{R}$ be an $L$-lipschitz function. Now, for all $x \in \mathbb{R}^{n}$, define $$\tilde{f}(x) = \left\{ \begin{array}{cc} \|x\|_2 f\left(\frac{x}{\|x\|_2}\right) & x \neq 0\\ 0 & x = 0 \end{array}\right. $$ where…
WazyMaze
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Finding the smallest lipschitz constant

Given is $f(x_1,x_2) = \sqrt {(6x_1)^2 + (9x_2)^2}$ Calculate the smallest Lipschitz constant: $L > 0$ so that: $ |f(x)-f(y)| \leq L||x-y|| $ for all $x,y\in {\Bbb R}^2$ and $||.||$ being the Euclidean norm. So far the exercises I've been doing have…
tictac
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Lipschitz condition and lipschitz continuity.

Is there a difference between the Lipschitz condition with Lipschitz continuity? What is that difference if there is the difference?
Rose
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Is this function involving indicator function Lipschitz?

Is this function $$x1_{\{x>0\}}(x)$$ Lipschitz? It's not differential so mean value cant be used here.
Vaolter
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Construct indifferentiable Lipschitz function

Construct a Lipschitz function on the whole real line, which is not differentiable at exactly 3 points.
rrz.math
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Example of Lipschitz contraction?

Can you please give me some example of Lipschitz contraction that is easy to visualize? Do I understand it right that in R the K=1 (Lipschitz constant) describe only the constant functions? Is it possible to have K<1 on R?
Keloo
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If $f(x)$ is Lipschitz, then is $xf(x)$ Lipschitz?

Suppose $f(x)$ is Lipschitz (globally or locally), what can we say about the product $xf(x)$. Is it also Lipschitz?
Fraïssé
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Relationship between and a Lipschitz function and a derivative of that.

From my textbook, Measure and Integral, a function $f$ defined on $[a, b]$ is said to satisfy a Lipschitz condition on $[a, b]$ or to be a Lipschitz function on $[a, b]$, if there is a constant $C$ such that $$\left|f(x)-f(y)\right|\le C|x-y| \quad…
Danny_Kim
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