# 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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### Show that $f_n(x):=\sqrt[n]{x}$ is Lipschitz continuous and determine the Lipschitz constants $L_n$

The question: Let $a>0, n\:\epsilon\: \mathbb{N}$ and let $f_n:[a,\infty)\rightarrow \mathbb{R}$ be defined by $f_n(x):=\sqrt[n]{x}$. Show that $f_n$ is Lipschitz continuous and determine the ...
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### Questions on the Kantorovich-Rubinstein duality

Let $\mu,\nu$ be probability measures on a metric space $(E,d)$ endowed with the Borel $\sigma$-algebra and $$\operatorname W_d(\mu,\nu):=\inf_{\gamma\in\mathcal C(\mu,\:\nu)}\int d\:{\rm d}\gamma,$$ ...
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### Proof of existence and uniqueness of solution under control

I read a research paper in which a replicator dynamics ODE is considered: $\dot{x}_{i}(t)=\delta x_{i}(t)[\pi(i, \mathbf{x}(t), \mathbf{r}(t))-\pi(\mathbf{x}(t), \mathbf{x}(t), \mathbf{r}(t))]$ To ...
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In general, we know that for strongly convex functions for which we can compute the Hessian and find the Lipschitz constant $L$ of the gradient, gradient descent will converge provided that the step ...
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### When is the region between two Lipschitz graphs a Lipschitz domain?

For $f,g : \mathbb{R}^d \to \mathbb{R}$ Lipschitz with $L_1$ and $L_2$ norms, let $\Omega= \{ (a,b) \in \mathbb{R^{n+1}} : f(a) \leq b \leq g(a) \}$. I was wondering under which (hopefully mild) ...
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### Derivative has Linear Growth Implies Lipshitz

Let $f\in C^\infty(\mathbb{R}^d)$. If $f$ has linear growth i.e $$|\nabla f(x)|\leq C(|x|+1)$$ then is $f$ Lipshitz? attempt at proof : by Mean Value Theorem there exists $c\in (0,1)$ such that ...
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### If a function is Lipschitz, and differentiable, is its gradient also Lipschitz?

If $f(x)$ is Lipschitz, i.e. $$||f(x) - f(y)|| \le L||x-y||$$ is it's gradient also Lipschitz? $$||\nabla f(x) - \nabla f(y)|| \le K||x - y||$$ And does $L = K$ ?
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### A question about proof of Martingale Convergence Theorem. Why does the Uniform integrability imply the following fact?

Relying on the below definition of uniform integrability: Definition: A subset $\mathcal{U}$ of $\mathcal{L}^{1}$ is said to be a uniformly integrable collection of random variables if \begin{...
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### Verhulst model and Lipschitz dependancy

I have a differential equation as follow which is Verhulst model: $$I'(t) = \beta I(t)\left(1-\dfrac {I(t)}N \right)$$ So I wanted to see just if there is a solution to this equation and if it is ...
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### Contraction Proof - Prove that $d_H(S(A), S(B)) ≤ cd_H(A, B)$

I need a little help with this simple problem whilst revising; Let S be a contraction on D with constant 0 < c < 1 and A, B ∈ S. Prove that $d_H(S(A), S(B)) ≤ cd_H(A, B)$ where $d_H$ is the ...
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### Show that if $F(f(x))$ is AC on $[a,b]$, for all $f$ that are AC on $[a,b]$, then $F$ is lipshitz

I have no clue how to prove this. I am not sure if duplicates are allowed but no one answered it here: $F$ is Lipschitzian if for every $f$ AC，$F◦f$ is AC for 3 years. It is clear $F$ is AC by ...
### A question in the proof that Lipchitz continuous functions implies $W^{1,\infty}$. [duplicate]
In $\S 5.8.2$ of Evan's PDE book, there is a theorem about characterization of $W^{1,\infty}$. Here it says On the other hand assume now $u$ is Lipschitz continuous; we must prove that $u$ has ...