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 $y$.

Filter by
Sorted by
Tagged with
1
vote
0answers
11 views

Are the Fréchet differentiable functions a closed subspace of this normed space of Lipschitz functions?

Let $E$ be a $\mathbb R$-Banach space, $d$ be a metric on $E$ and $\mu$ be a probability measure on $(E,\mathcal B(E))$ with $$\int d(\;\cdot\;,0)\:{\rm d}\mu<\infty\tag1.$$ Moreover, let $$|f|_{\...
0
votes
0answers
17 views

A convex function on an open interval is continuous.

I'd like to show that A convex function defined on open interval is continuous. I get the result, f is Lipschitz function on any closed subinterval. But, I am stucked here. How can i get the ...
0
votes
1answer
17 views

Show that $C^\infty(\overline{\Omega}) \subseteq C^{0,1} (\overline{\Omega})$

Let $\Omega$ be a bounded, connected, open domain in $\mathbb{R}^d$ with smooth boundary. Denote by $C^{0,1} (\overline{\Omega})$ the space of continuous functions $u$ on $\overline{\Omega}$ such ...
0
votes
0answers
12 views

If $f$ is Fréchet differentiable, bound the Lipschitz seminorm by the supremum of the norms of $f$ and ${\rm D}f$

Let $E$ be a $\mathbb R$-Banach space, $$d(x,y):=\min(1,\left\|x-y\right\|_E)\;\;\;\text{for }x,y\in E,$$ $\Omega\subseteq E$ be open, $$|f|_{\operatorname{Lip}(d)}:=\sup_{\substack{x,\:y\:\in\:\Omega\...
0
votes
1answer
21 views

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 ...
0
votes
0answers
16 views

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 $$\operatorname W_d(\mu,\nu):=\inf_{\gamma\in\mathcal C(\mu,\:\nu)}\int d\:{\rm d}\gamma,$$ ...
0
votes
0answers
21 views

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 ...
0
votes
0answers
26 views

Gradient descent with non-Lipschitz continuous gradients

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 ...
0
votes
0answers
10 views

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) ...
1
vote
1answer
14 views

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 ...
0
votes
1answer
30 views

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 $L = K$ ?
0
votes
0answers
40 views

Show that if a function $f$ Lipschitz continuous on $X$, $f$ has to be uniformly continuous on $X$.

Show that if a function $f$ is Lipschitz continuous on $X$, $f$ has to be uniformly continuous on $X$. My attempt: (1) The definition of Lipschitz continuity for $f$ on $X$ is: $\exists L \in \...
2
votes
2answers
29 views

Expression for Lipschitz constant for the $L^p$ norm function on $\mathbb R^n$

Let $p \ge 1$ and $f:\mathbb R^n \to \mathbb R$ is given by $f(x):= \|x\|_p.$ Then is $f$ a Lipschitz function, and if yes, what's its Lipschitz constant? For $p=1,$ I see that it's $\sqrt{n}$ ...
5
votes
1answer
34 views

Cantor–Schröder–Bernstein for Lipschitz maps?

Let $X,Y$ be metric spaces and suppose there exist bijective Lipschitz functions $f : X \rightarrow Y$ and $g : Y \rightarrow X$. Does there necessarily exist a bijective bi-Lipschitz function $h : X \...
0
votes
0answers
12 views

Bound on the third derivative with Lipschitz condition

I'm trying to understand the following paragraph from Boyd & Vandenberghe, page 488: (...) we assume that the Hessian of $f$ is Lipschitz continuous on $S$ with constant $L$, i.e., $$ \| \...
1
vote
0answers
26 views

Lipschitz maps cannot increase the volume of a Borel set by a factor greater than $k^n$.

Let $(M,\,g,\,d,\,vol_g)$ be a Riemannian manifold with metric $g$, geodesic distance $d$ and volume form measure $vol_g = \sqrt{\det(g_{ij})}\cdot m$ ($m$ = Lebesgue measure) and a Lipschitz map $f:K\...
0
votes
1answer
27 views

Locally Lipschitz? Globally Lipschitz

Does this function $$f(x,y) = -\frac{2xy}{(\frac{3}{2}\sqrt{|y|}+1+x^2)} $$ Satisfy the local Lipshitz condition $$|f(x,y_2)-f(x,y_1)| \leq M|y_2-y_1| $$ for $x,y \in \mathbb{R}$ I have no idea how ...
0
votes
0answers
24 views

How would I prove (or disprove) $f(x)=\frac{1}{x}$ is not a contraction on $([1,\infty),|.|)$?

How would I prove (or disprove) $f(x)=\frac{1}{x}$ is not a contraction on $([1,\infty),|.|)$? So I computed the derivative of $f(x)$ and realised that the modulus of the derivative is $1$ at $x=1$ ...
0
votes
1answer
67 views

Uniqueness of O.D.E

Prove that the o.d.e. $(\frac{3}{2}\sqrt{|y|}+1+x^2)\frac{dy}{dx}+2xy=0$ has unique local solutions with $y(x_0) = y_0$ for any $x_0$ and $y_0$. Does the existence and uniqueness theorem for o.d.e's ...
0
votes
3answers
27 views

Continuous function on $[0,1]\to [0,1]$ that is not Lipschitz Continuous?

Continuous function on $[0,1]\to [0,1]$ that is not Lipschitz Continuous? One example I could perhaps think of is $f(x)=sin(\frac{1}{x})$ where we define $f(0)=0$. Then this function has the ...
4
votes
2answers
55 views

Is a continuous function bounded by Lipschitz continuous function also Lipschitz continuous?

Is a continuous function bounded by Lipschitz continuous function also Lipschitz continuous? For example this function $$f(x,y)=\frac{x^3-3xy^2}{x^2+y^2}$$ and $f(0,0)=0$. I can show that $$\left|...
2
votes
1answer
29 views

Convex conjugate: Lipschitz continuity of the argmax function

Let $s,\delta\in\mathbb{R}^{N}$, $S\subseteq\mathbb{R}^{N}$ be a compact convex set, $f:S\rightarrow\mathbb{R}$ be a twice differentiable strictly convex function on $S$ and $$s\left(\delta\right)=\...
2
votes
1answer
21 views

Showing that a Hölder map is also Hölder with a smaller Hölder parameter

Let $X\subset{\mathbb{R}^d}$ be bounded and let $f:X\rightarrow{}\mathbb{R}^d$ be an $\alpha$-Hölder map, that is to say: $$\|f(x)-f(y)\|_2\le{}C\|x-y\|_2^{\alpha},\text{ for some }\alpha>0\text{ ...
1
vote
1answer
85 views

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{...
0
votes
1answer
46 views

Prove that f is a lipschitz function if $\exists K \in \mathbb{R^+} \forall x,y \in \mathbb{R}: |f(y)-f(x)| \le K|\cos y - \cos x|$

I'm trying to prove that $f : \mathbb{R} \rightarrow \mathbb{R} \phantom{2}$ is a Lipschitz function if: $\exists K \in \mathbb{R^+} \phantom{1}\forall x,y \in \mathbb{R}: \lvert f(y)-f(x) \rvert \...
-2
votes
1answer
17 views

About any estimation of the quantity $|f(x)-x|$ [closed]

Let $f:(0,1)→(a,b)$ be a 1-Lipschitz function where $b>0$. I am asking about any estimation of the quantity $|f(x)-x|$.
0
votes
1answer
41 views

How can I test the Lipschitz condition for a second order system?

I have the second order autonomous system $$x_1 '= ax_1 - bx_1x_2 \\ x_2 '= -cx_2 + dx_1x_2$$ How can I apply the Lipschitz condition to this second order system? Can anyone explain step by step ...
0
votes
0answers
21 views

Does Lipschitz continuity of $f$ and $(f|_{\text{dense set}})'=0$ implies $f$ is a constant function?

I'm a student studying mathematical analysis, and as in this post, I want to find a [condition] such that if $A \subseteq \mathbb{R}$ is dense and $f:A \to \mathbb{R}$ satisfies [condition], then $f$ ...
0
votes
1answer
42 views

Lipschitz constant of continuous and piecewise linear functions

I want to calculate the Lipschitz constant of a continuous and piecewise linear function $f:[0,1]^2\rightarrow R$, like this \begin{equation*} f(x_1,x_2)=\left\{ \begin{aligned} 2x_1+x_2, &\quad\...
2
votes
0answers
36 views

Finding a constant in inequality for softmax function

Let $x, y$ be vectors in $\mathbb{R}^n$, and $\textrm{softmax}: \mathbb{R}^n \to \mathbb{R}^n$ is a vector function such that: $$\textrm{softmax}(x)_i = \frac{e^{x_i}}{\sum^n_{j=1} e^{x_j}}.$$ What ...
2
votes
4answers
219 views

Prove that, for each $n\in\mathbb{N}$, $f_n(z)=|z|e^{in\arg(z)}$ is Lipschitz with constant n.

Suppose $n\in\mathbb{N}$ and $f_n:(\mathbb{C},|\cdot|)\rightarrow(\mathbb{C},|\cdot|)$ is defined via the map $z\mapsto |z|e^{in\arg(z)}$. How do you show that $f_n$ is a Lipschitz map with Lipschitz ...
0
votes
1answer
45 views

Differentiability of a Lipschitz function in one point [closed]

Let $f:\mathbb{R}^{2}\to\mathbb{R}$, such that $f$ is $1$-Lipschitz. I assume that $\frac{\partial f}{\partial x}(0,0)$ exists and $\frac{\partial f}{\partial x}(0,0)=1$. Does it imply that $f$ is ...
2
votes
1answer
58 views

Can gradient of a function be Lipschitz continuous but the function is not Lipschitz?

Can gradient of a function be Lipschitz continuous but the function is not Lipschitz? Any figurative example where gradient is Lipschitz but the function is not Lipschitz possible would be really ...
0
votes
0answers
21 views

Are all functions with Lipschitz continuous gradient also strongly convex?

I am sorry for asking probably well known question. But I am so confused. Are all functions, say $f$, with Lipschitz continuous gradient, say, $\nabla f$ is $L$-Lipschitz, also strongly convex ...
0
votes
1answer
32 views

How to prove the 1 Lipschitz function defined on the closed unit ball has a fixed point

I need to prove that the 1 Lipschitz function has a fixed point: $\|f(x)-f(y)\|≤ \|x-y\|$ for all $x,y\in B$, where $B$ is the closed unit ball in the $R^n$. I want to apply the contraction mapping ...
0
votes
1answer
34 views

Real Analysis on Functions with Lipschitz Derivatives

I saw a wired argument somewhere and thought maybe this would be a good practice and somehow I can't wrap my head around it: Let $f(x)$ be Lipschitz and have Lipschitz derivatives everywhere. If we ...
0
votes
0answers
19 views

Is $\min$ a non-expansive function?

I came across a note where it does something like the following: \begin{align} l &= \min \{H, \max_{j\in [M]}\phi^\top(\theta + \xi^{j}) \} - \delta \\ & \geq [\max_{j\in [M]}\phi^\...
1
vote
1answer
44 views

Inequality for convex function say $f$ with $L$-Lipschitz continuous gradient: $( x - y)^T \left( \alpha \nabla f(x) - \beta \nabla f(y)\right)$?

Any known known bound for a convex function say $f$ with $L$-Lipschitz continuous gradient (for $\alpha, \beta \in \mathbb{R}$, which can be $\alpha \neq \beta$): $( x - y)^T \left( {\color{red} \...
0
votes
0answers
14 views

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 ...
0
votes
0answers
13 views

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 ...
1
vote
0answers
21 views

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 ...
0
votes
0answers
10 views

Lipschitz continuity of multivariable function in expected value

Suppose $h:\mathcal{X \times Y \times W}\rightarrow\mathcal{X}$, with $\mathcal{X,Y,W} \subseteq \mathbb R^d$, $d \in \mathbb N$ is Lipschitz in $x,y$, i.e., $$ \| h(x,y,w) - h(x',y',w) \|_2 \le L_h (...
0
votes
1answer
21 views

Using Picards theorem to show that the initial value problem has a unique solution

I am trying to show that the IVP $$x'=\sqrt{x(t)}+1, t\in[0,1],\\x(0)=0, (t_0=0)$$ has a unique solution and show whether the initial value problem satisfies the assumptions of Picard’s Theorem, ...
0
votes
1answer
70 views

Condition Number and Lipschitz Constant

I'm interested in mathematical optimization and have been reading about the concepts of Condition Number and Lipschitz Continuity. They seem very related to me. Is the absolute condition number of a ...
-2
votes
2answers
31 views

Show that the function $f$ is a contraction

I want to show that the function $f:\mathbb{R}\rightarrow \mathbb{R}$ with \begin{equation*}f(x)=\frac{1}{2}\sin x-1\end{equation*} is a contraction. A function $g : \Omega \rightarrow \Omega$ is a ...
6
votes
1answer
85 views

Any known bounds for convex function say $f$ with $L$-Lipschitz continuous gradient: $( x - y)^T A \left( \nabla f(x) - \nabla f(y)\right)$?

There are several known bounds for a convex function say $f$ with $L$-Lipschitz continuous gradient, for instance, \begin{align} \left( x - y \right)^T \left( \nabla f(x) - \nabla f(y)\right) \leq ...
1
vote
0answers
31 views

Vector-valued continuously differentiable function of several variables

My lecture notes on ODEs state the following in the section on unique solutions: Definition: Let $I \subset \mathbb{R}$ be an interval and let $D \subseteq I \times \mathbb{R}^d$ be open. Then $f : ...
1
vote
1answer
40 views

Lipschitz function and uniform convergence

I am struggling with the proof of the following theorem: Let $(f_n)$ be a sequence of differentiable functions in a closed and bounded set $[a,b]$ s.t. $(f_n(x))$ is convergent for each $x\in [a,b]$ ...
1
vote
1answer
30 views

Existence theorem for the SIR epidemic differential equations

Well these days, everybody is talking about the epidemic model SIR which is given by the following differential equation: $$ \begin{cases} S'(t)=-aS(t)I(t)\\ I'(t)=aS(t)I(t)-bI(t)\\ R'(t)=bI(t)\\ S(0)...
1
vote
1answer
43 views

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 ...

1
2 3 4 5
24