# 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 $y$.

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### Uniformly Continuous and locally Lipschitz but not Globally Lipschitz Function on a "Connected but not compact" set

I know such a function exists but I can’t find an example. I have the famous example f:]0,inf[ --->R f(x)=sqrt(x) function. But I can't find any other function which is Uniformly Continuous and ...
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### Prove that a certain function is 2-lipschitzienne

I'm trying to prove the following proposition. Prove that g:E×E→ℝ+, (x,y)→d(x,y) is a 2-lipschitz function. The distance on ℝ+ is the usual distance The distance on E×E is defined as d'((x,y),(x',y'))=...
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### How to bound scalarized gradient difference norm in terms of smoothness in convex optimization?

We know if a convex function is $\mu$-smooth, the following inequality is true: $\| \nabla g (u) - \nabla g(v) \| \leq \mu \|u-v\|$ I want to derive an bound for the following slightly different term ...
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### An integral inequality with lipschitz function

Suppose that $m=\min_{x\in[a,b]}f(x),K=\int_a^b\frac{dx}{f(x)}$, $f(x)>0$ and it satisfies $\forall x,y\in[a,b],|f(x)-f(y)|\le L|x-y|$. Prove that $\int_a^bf(x)dx\le \frac{e^{2LK}-1}{2L}m^2$. I've ...
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### Convergence of $C^2$ functions with fixed point iteration

In Ridgway Scott's Numerical Analysis, the following two problems appear: In the first exercise, there was a clear mention of $x_0$ being sufficiently close to $\alpha$. So I knew that I should be ...
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I am interested in the smallest coordinate of a vector. Given a positive real vector $x=(x_1,...,x_n)$, I need your help to determine if the function $f(x)= min_{1\leq i \leq n}x_i$ is Lipschitz. If ...
### Showing a function $\mathbb R \to \mathbb R^2$ is bilipschitz
Problem: Is the function $f\colon \mathbb R \to \mathbb R^2$, defined by $f(x)=(x, \cos x)$, $M$-bilipschitz for some $M\geq 1$? (Using the standard metric) So far: By the MVT we have \$\lvert \cos x -\...