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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 $y$.

1899 questions
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What is the intuition behind uniform continuity?

There’s another post asking for the motivation behind uniform continuity. I’m not a huge fan of it since the top-rated comment spoke about local and global interactions of information, and frankly I just did not get it. Playing with the definition,…
Spencer Kraisler
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Is a function Lipschitz if and only if its derivative is bounded?

Is the following statement true? Let $f: \mathbb{R}\to\mathbb{R}$ be continuous and differentiable. $f$ Lipschitz $\leftrightarrow \exists M:\forall x\in\mathbb{R}\ |f'(x)|\leq M$ If $f'$ is bounded, it is Lipschitz, that's obvious. Does that work…
JonTrav
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Why does a Lipschitz function $f:\mathbb{R}^d\to\mathbb{R}^d$ map measure zero sets to measure zero sets?

Why does a Lipschitz function $f:\mathbb{R}^d\to\mathbb{R}^d$ map measure zero sets to measure zero sets? It is easy to prove this statement if the domain is bounded. Is there any way to extend the argument to unbounded domains? Can anyone give me…
mysterious
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Lipschitz smoothness, strong convexity and the Hessian

I'm working with the following two concepts: Lipschitz Smoothness - a function $f$ is Lipschitz smooth with constant $L$ if its derivatives are Lipschitz continuous with constant $L$, in other words if for any $x$ and $y$, $$ \| \nabla f(x) -…
Daniel
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$\sqrt{x}$ isn't Lipschitz function

A function f such that $$ |f(x)-f(y)| \leq C|x-y| $$ for all $x$ and $y$, where $C$ is a constant independent of $x$ and $y$, is called a Lipschitz function show that $f(x)=\sqrt{x}\hspace{3mm} \forall x \in \mathbb{R_{+}}$ isn't Lipschitz…
Adam
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The definition of locally Lipschitz

Marsden's Elementary Classical Analysis seems to indicate this definition: A function $f:A{\subset}\mathbb R^n\to\mathbb R^m$ is locally Lipschitz if for each $x_0{\in}A,$ there exist constants $M{>}0$ and $\delta_0{>}0$ such that…
ryang
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A multivariate function with bounded partial derivatives is Lipschitz

I'm curious if I've done this correctly -- please offer suggestions/corrections if not! I'm new to working in $\Bbb R^n$ so clear insights would be appreciated. The problem: Let $f:\Bbb R^2 \to \Bbb R$ be such that each $D_1f$ and $D_2f$ are…
Newb
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Isometry in compact metric spaces

Why is the following true? If $(X,d)$ is a compact metric space and $f: X \rightarrow X$ is non-expansive (i.e $d(f(x),f(y)) \leq d(x,y)$) and surjective then $f$ is an isometry.
student
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A continuously differentiable map is locally Lipschitz

Let $f:\mathbb R^d \to \mathbb R^m$ be a map of class $C^1$. That is, $f$ is continuous and its derivative exists and is also continuous. Why is $f$ locally Lipschitz? Remark Such $f$ will not be globally Lipschitz in general, as the one-dimensional…
bass
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Lipschitz-constant gradient implies bounded eigenvalues on Hessian

I've read in a few places that if we have a Lipschitz gradient $$\|\nabla f(x) - \nabla f(y)\|\leq L\|x-y\|,\, \forall x,y, $$ we can equivalently say $\nabla^2f\preceq LI.$ But I'm having a hard time showing this. (Equivalently, I want to show $z^T…
Leland Stirner
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Is a Lipschitz function differentiable?

Is a Lipschitz function differentiable? I have been wondering whether or not this property applies to all functions. I do not need a formal proof, just the concept behind it. Let $f: [a,b] \to [c,d]$ be a continuous function (What is more - it…
Aemilius
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Understanding Lipschitz Continuity

I have heard of functions being Lipschitz Continuous several times in my classes yet I have never really seemed to understand exactly what this concept really is. Here is the definition. $\left | f(x_{1})-f(x_{2}) \right |\leq K\left | x_{1}-x_{2}…
Erock Brox
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Continuous differentiability implies Lipschitz continuity

Here's a statement in Zygmund's Measure and Integral on page 17: If $f$ has a continuous derivative on $[a,b]$, then (by the mean-value theorem) $f$ satisfies a Lipschitz condition on $[a,b]$. This does not seem obvious to me. How can I show…
1LiterTears
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Is Lipschitz's condition necessary for existence of unique solution of an I.V.P.?

Is Lipschitz's condition necessary condition or sufficient condition for existence of unique solution of an Initial Value Problem ? I saw in a book that it is sufficient condition. But I want an example to prove it sufficient. That is I want an…
Empty
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Lipschitz implies bounded gradient

Assume $f:\mathbb{R}^n \to \mathbb{R}$ is convex, and $L$-Lipschitz, so $|f(x)-f(y)|\leq L\|x-y\|$. I would like to show that $\|\nabla f(x)\|\leq L$. In one dimension this is a straightforward consequence of the fact that convexity implies…
guest01645
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