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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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g is a Lipschitz on the interval [α-A,α+A] for some A>0. Prove |g'(α)|≤λ

Suppose that g is a C1 function such that the Lipschitz estimate |g(x)-g(y)|≤λ|x-y| holds an interval [α-A,α+A] for some A>0. Prove that |g'(α)|≤λ. (Hint: consider the difference quotients used to define g'(α).)
Bob R.
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Let $f(x,y)=2x-4\frac{y}{x}$, $f(0,0)=0$. Show that $f$ is not Lipschitz.

Let \begin{align} f(x,y)&=2x-4\frac{y}{x} \\ f(0,0)&=0, \end{align} for $|x|\leq 1$, $0\leq y\leq x^2$. Show that $f$ is not Lipschitz. I calculated its derivative and it is unbounded, but I don't know hot to prove it by definition since…
José
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lipschitz continuity from a constant c?

Let $f : [1,\infty) \to \mathbb R$ be uniformly continuous. Show that there exists a number $C > 0$ such that $$|f(x)|\le Cx$$ for all $x \ge 1$.
user238168
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Find a Lipschitz constant for a quadratic function restricted to a ball

Fix $a$ in ${\mathbb R}^d$ and $b$ in ${\mathbb R}$, consider $x$ in ${\mathbb R}^d$ where $\|x\| \le B$. Is the following function Lipschitz? (and if so, what is a Lipschitz constant?) $$f(x) = (\langle a, x \rangle + b)^2$$
Xi Wu
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Checking a function is Lipschitz or not

I am learning differential equation and the uniqueness theorem for 1st order ODE said that if $y' = f(x,y)$ where $f(x,y)$ is Lipschitz w.r.t. $y$ or $\frac{df}{dy}$ is continuous, then the ODE has at most one solution. But my question is not about…
Nighty
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Prove function do not satisfy Lipschitz condition

How to show that for function $f:\mathbb{R}^n\mapsto\mathbb{R}$ $f(\mathbf x)=||\mathbf x||^{3/2}$, the Lipschitz condition $||\nabla f(x)-\nabla f(y)||\le L||x-y||$ for all $x,\, y$ is not satisfied for any $L$?
lolibility
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Existential theorem - Lipshchitz

I have to find out whether the equation $$y' = \sqrt{y} $$ by the condition $$y(1) = 1 $$ satisfies the assumptions of the Existential theorem. Now I claim that the function is continuous around x = 1, which means then we have one exact solution for…
Zenga
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Proof of Lip(F) being the maximum of norm of the gradient of F(x) for x in the domain

I am working through the proof of finding the lipschitz of a continuously differentiable function. Specifically, trying to prove that it is equal to $Lip(F) = \max \{ \lVert \nabla F(x) \rVert | x \in \Omega \} $ I am able to prove that $Lip(F) \leq…
Prabhjot Singh Rai
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Is the divergence of positive definite matrix function locally Lipschitz?

Let $a:\mathbb{R}^d\to S_d^{++}$ where $S_d^{++}$ is the set of $d\times d$ positive definite matrices. Suppose that $a$ is $C^1$. Then by a theorem of Phillips and Sarason (Rogers and Williams Theorem 12.12., p.134), we know that $\sigma(x) =…
Nap D. Lover
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Is the function $f(x, y)= x\sin y+y\cos x $ Lipschitz on $[-a, a]\times[-b,b]?$

My attempt Let $(x, y_1)$ and $(x, y_2)$ be two points in the rectangle $R=[-a, a]\times[-b,b].$ Then $$|f(x, y_1)- f(x, y_2)|\le|x||\sin y_1 -\sin y_2| + |y_1-y_2|$$ I can't understand how to proceed further; please give me some hints
Old
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Does Lipschitz derivative imply uniform differentiability of $f$ (defined on open interval)?

Let $f: (0; 1) \rightarrow \mathbb{E}^n$. Does $f'$ being Lipschitz imply that $f$ is uniformly differentiable? In my professor's textbook (yet unreviewed professionally, thus I'm coming here for an answer) this implication is a part of a proof of a…
michael15346
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How to show that L2 norm is Lipschitz if the input domain is bounded?

I guess that $||w||_2^2$ is 2B Lipschitz when $||w||
geradism
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How do I prove that $f : \Bbb{R}\setminus [2,4] → \Bbb{R}$ such that $x\mapsto \frac{1}{x-3}$ is Lipschitz continous?

It sounds simple enough, but i how do i work with the interval? Do I have to apply a case distinction for $> [2,4]$ and $< [2,4]$?
Sungoku
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Lipschitz constant of a function on a compact set

Let $K$ be a compact set of a normed space $(X, \|\cdot\|)$. Suppose $f \colon K \to \mathbb R$ satisfies $$ \frac{|f(x) - f(y)|}{\|x-y\|} < 1 \qquad \text{for all } x, y \in K \text{ with $x \neq y$}. $$ Then $f$ is Lipschitz continuous with…
Keba
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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: "$f$ is locally Lipschitz in $y$ uniformly in…
Andrew
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