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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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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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Lipschitz constant tending to zero implies function is constant

Let $f$ be a Lipschitz function defined on a ball $B$ in $\mathbb{R}^n$, and assume that the following property on the Lipschitz constant (restricted to smaller balls) holds: $$ \lim_{r\to 0}\sup_{|x-y|\leq r} \frac{|f(x)-f(y)|}{|x-y|}=0. $$ Can we…
MathFreak
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Justification of inequality

Denote $$L(f) = \sup_{p,q \in X}{\dfrac{\rho(f(p),f(q))}{\rho(p,q)}}$$ The following proposition is extracted from the book 'Lipschitz Algebras' by Weaver. Proposition $1.2.4$ Let $X$ and $Y$ be metric spaces and let $f$ and $f_i (i \in I)$ be…
Idonknow
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Does the Lipschitz constant depend on the norm?

Does the Lipschitz constant depend on the used norm? In my courses I know that the Lipschitz constant does not depend on the norm in $\mathbb{R^n}$ because it is complete so all the norms are equivalent. I found nothing in the online litterature…
Conjecture
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Lipschitz Function over the real axis

Let the following function: \begin{equation} \pi(x) = \frac{e^x}{1 + e^x},\quad \textrm{for all } x \in \mathbb{R}. \end{equation} I want to know if it is Lipschitz? Here my proof and I like to know if it is true? \begin{equation} \pi'(x) =…
mathdatastatsml
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Smallest Lipschitz constant of gradient of a function

I have to show that $2\Vert{A}\Vert$ is the smallest Lipschitz constant of $\nabla f$, where $f(x)=x^{T}Ax+2b^{T}x+c$. $\forall x,y\in \mathbb{R}^n, \Vert\nabla f(x)-\nabla f(y)\Vert = \Vert2(Ax+b)-2(Ay+b)\Vert = \Vert2A(x-y)\Vert \le \Vert2A\Vert…
Mina
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The exponential function is locally Lipschitz continuous with the Lipschitz constant $K=1$

The exponential function $x→e^{x}$ becomes arbitrarily steep as $x → ∞$, and therefore is not globally Lipschitz continuous, despite being an analytic function. I am asking if it is possible to find certain regions for the variable $x$ in which the…
Safwane
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Correct way to show that a Lipschitz condition is satisfied

In one of my homework problems, I have an IVP $$ y'=e^{t-y}, \hspace{5mm}where\hspace{3mm}0≤t≤1,\hspace{3mm}y(0)=1 $$ And I need to show that $$ f(t,y)=e^{t-y} $$ satisfies a Lipshitz condition. To do this, I used the method shown in class. Same as…
Alex Wolski
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Max operator over a lipschitz function is still lipschitz?

I have a function $f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ which is $L$-Lipschitz (over the Manhattan distance). Is $\max_y f(x,y)$ still Lipschitz? In my opinion, it is true, and I can prove it, but I'm not sure whether the proof is correct or…
Sam
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Lipschitz property from boundedness of a function and its partial derivatives

Suppose the functions $h(t,x)$ and $f(t,x,z,\epsilon)$ and their partial derivatives up to the second order are bounded in some compact domains that contain the origin, where $\epsilon \in [0,\epsilon ^∗)$, $\epsilon^*<<1$. Also suppose that…
M3053
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How do we conduct the infimum of $inf_y()+⟨∇(),−⟩+\frac{L}{2}‖−‖^2$ in the quadratic upper bound

I am trying to understand the Consequence of quadratic upper bound in the 16th slide of UCLA ese236c. The left-hand side of the inequity is still confusing to me. I know in this similar question, someone gave an explanation, but he concluded the…
Zihao
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Is $\frac{u}{|u|}$ 1- Lipschitz?

I want to prove that the application $\{s\in \mathbb{R}^3:|s|\geq 1\} \to S^2 ,u\mapsto \frac{u}{|u|}$ is Lipschitz with constant $1$. If I add ($\frac{u|u|-u|u}{|u||v|}|$), I find the constant equals $2$ ( the norm used is the euclidien norm). Is…
Romi
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Locally Lipschitz and Lipschitz continuous on compact sets are equivalent

By Wikipedia's definition, a mapping $f:X\mapsto Y$ ($X,Y$ are metric spaces) is locally Lipschitz if for every $x\in X$ there exists a neighborhood $U$ of $x$ such that $f$ restricted to $U$ is Lipschitz. It further claims that if $X$ is locally…
Andrew Yuan
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Lipschitz Condition for Vector Valued Functions

While understanding the concept of Existence and Uniqueness Theorem for Initial Value Problems of System of Ordinary Differential Equations, I first came across "vector - valued functions" and the condition for the uniqueness of solution in vector…
Aniruddha Deshmukh
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absolute value of 2 complex exponentials

Sorry if this is a dumb question. Trying to understand what this simplifies to: $$f(t) = |e^{iwt} - e^{iwv}|$$ I don't understand how to get a nice numerical answer to this as the exponentials use different variables to one another... Its related…
CN railfan
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