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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 $y$.

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21 views

Showing function is not Lipschitz continuous

For an analysis exercise, I had to show that the function $\sqrt{1-x^2}$ was uniformly continuous, but not lipschitz continuous on the interval $[-1,1]$. I was able to show it was uniformly continuous,...
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15 views

Non expansive function

Let $F:D \subset X \rightarrow X$ being X a Banach space and $I + \lambda F $ onto for $\lambda >0$ then $R_{\lambda}= (I + \lambda F)^{-1}$ is no expansive. How can i prove this?.
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22 views

If $f$ has an unbound derivative - is $f$ necessarily non-Lipshitz?

So we assume that $f$ is differentiable on some closed interval $[a,b]$ and has a derivative $f'$ which is not bounded. Is there a way to show that $f$ is non-Lipshitz? If not, is there some ...
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1answer
37 views

Showing $\{f(\frac{1}{n+1})\}$ converges in $\mathbb{R}$

Question: Let $f:(0,1) \to \mathbb{R}$ be a differentiable function such that $|f'(x)| \leq 5$, for all $x \in (0,1)$. Show that the sequence $\{f(\frac{1}{n+1})\}$ converges in $\mathbb{R}$. ...
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1answer
24 views

Prove convolution of (f - Lipschitz, bounded) and some (g $\in L^1$) fulfills the Lipschitz criterion [on hold]

Prove that the convolution of bounded, Lipschitz-continuous function f : $\mathbb R^n \rightarrow \mathbb R$ and function $g \in L^1(\mathbb R^n)$ fulfills the Lipschitz criterion itself.
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1answer
8 views

Deriving an inequality for a set of functions that have equi-Lipschitz first derivatives.

let $ F =\{ F_t \}_{t \in \mathbb{N}}$ be a sequence of continuously differentiable real valued functions such that the sequence $ F' =\{ F'_t \}_{t \in \mathbb{N}}$ is equi-Lipschitz , i.e. there ...
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1answer
57 views

Lipschitz continuity of $e^{\sin}$

We want to use the Picard-Lindelöf-Theorem to show that the ODE $$y'=\mathrm{e}^{\sin(ty)}$$ has a unique solution on $\mathbb{R}$ with the initial value $y(0)=0$. As far as I know, we have to ...
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0answers
23 views

Weak differentiability vs differentiability almost everywhere

Is there any relation between weak differentiability and differentiability almost everywhere? Does one imply the other? Does Lipschitz continuity imply both of them? Thank you.
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2answers
118 views

Prove or disprove that there exists $K$ such that $|f(x)-f(y)|\leq K |x-y|,\;\forall\;\;x,y\in[0,1],$ edited version.

Let $f$ be a function on $[0,1]$ into $\Bbb{R}$. Suppose that if $x\in[0,1],$ there exists $K_x$ such that \begin{align}|f(x)-f(y)|\leq K_x |x-y|,\;\;\forall\;\;y\in[0,1].\end{align} Prove or disprove ...
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2answers
51 views

Prove that $f(x)=x\sin(1/x)$ for $x\ne0$, $f(0)=0$, is not Lipschitz on $[0,1]$

Prove that $f(x)=\cases{0& if $x=0$\\x\sin(1/x)& otherwise,}$ is not Lipschitz on $[0,1]$ MY TRIAL My idea is to show that $f$ does not have a bounded derivative. So, suppose for ...
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1answer
18 views

Showing that the system of a planar curve is Lipschitz in y

This question below, which was given in an exam, goes as follows: A planar curve $y(x)$ is such that its curvature, $$k(x)=y''(x)/(1+y'(x)^2)^{3/2}$$ is equal to its height $y(x)$. Write the ...
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2answers
35 views

How can $\ln(x+2)$ have a fixed point in $(-2,-1]$?

I have to determine the fixed points of $\ln(x+2)$. So as a first step I plotted $\ln(x+2)-x$ and found that it does have two fixed points. One between $[0, \infty]$, which is perfectly fine and one ...
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1answer
36 views

Example of a nowhere differentiable contraction mapping

The Weierstrass function https://en.wikipedia.org/wiki/Weierstrass_function is a pathological example of a continuous nowhere differentiable function.. Since a conttaction mapping is necessarily ...
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1answer
66 views

Lipschitz constant for a second order nonlinear differential equation

I'm trying to calculate the Lipschitz constant for a second order nonlinear differential equation: $$y'' + y' + y^n = 0, \; y(0)=0, \; y'(0)= 0 \text{ and } n>1$$ Should I solve for $y(x)$ and ...
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2answers
43 views

A Lipschitz function must has a set of intervals s.t. $\cup_kI_k=\mathbb R$ and $f$ is either convex or concave on either $I_k$?

Let $f:\mathbb R\to\mathbb R$ be a continuous function. $f$ is said to be "regular" if there exists a set of intervals $\{I_k\}_{k\in K}$ (indexed with $k$ where $K$ is an arbitrary set), such that $...
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1answer
71 views

Can a Lipschitz continuous function be linear almost everywhere but not linear everywhere? [closed]

Can a Lipschitz continuous function be linear almost everywhere but not linear everywhere? (:sorry for ambiguity) The almost everywhere here is defined as: Let $f:\mathbb R^k\to\mathbb R$. $\nabla f(...
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1answer
44 views

Do strongly equivalent metrics have the same covering numbers?

Let $(X,d)$ be a metric space. Then for any $r>0$, the $r$-covering number of $X$ is the minimum number of open balls of radius $r$ needed to cover $X$. And if $d_1$ and $d_2$ be two metrics on ...
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1answer
65 views

If $f$ is $C^{1}$ and $\sup_{x \in \mathbb{R}}|f'(x)| = \infty$ , prove that $f$ cannot be uniformly continuous

I'm looking for help with a proof that I still cannot figure it out. Here is the statement: "If $f:\mathbb{R} \rightarrow \mathbb{R}$ is $C^{1}$ and $\sup_{x \in \mathbb{R}}|f'(x)| = \infty$ , prove ...
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1answer
38 views

Are uniformly equivalent metrics with the same bounded sets strongly equivalent?

Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are uniformly equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ are ...
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0answers
29 views

How can I prove that $(f'(x))^2 \le 2Lf(x)?$ [duplicate]

How can I prove that if $f$ is function from $\Bbb R \to (0, +\infty)$ and for every $x$ and $y$ $$|f'(x)-f'(y)| \le L|x-y|,$$ ($L$ is not depend of $x$.) then for every $x$ $$(f'(x))^2 \le 2Lf(x)?...
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0answers
20 views

estimates for parameter dependent differential equation

Consider some function $f:\mathbb{R}\times X \times \Omega,$ $\Omega \subset X$ and the differential equation \begin{equation} \dot{x}(t)=f(t,x,p), \end{equation} where $p \in \Omega$ is some ...
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1answer
61 views

Prove Lipschitz condition

I'm working this function: $f(t,x)=\frac{4t^3x}{t^4+x^2}$ for $(t,x)\neq(0,0);$ $f(t,x)=0$ for $(t,x)=(0,0)$ As I proved continuity using the line $x=t^2$, I need to study if it's globally ...
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1answer
26 views

Metric on real line

Let $(X,d)$ be a metric space and fix some $x_{0}\in X$. Show that the function $f(x) = d(x_{0},x)$ is Lipschitz continuous? My problem wasn't with solving the question originally I did that using ...
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1answer
54 views

Lipschitz function and incomplete range

Find a Lipschitz function $f:\mathbb{R} \rightarrow \mathbb{R}$ that has incomplete range. What is incomplete range of a function? Does it has to do with completeness of a metric space? Any help ...
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1answer
45 views

Prove $f(x) = x^2\sin\left(\frac{1}{x}\right)$ is Lipschitz (no use of derivative)

Prove that $f:\mathbb{R}\to \mathbb{R}$ such that $$ f(x) = \left\{ \begin{array}{c l} x^2\, \sin\left(\frac{1}{x}\right) & ,\quad x\neq 0\\ 0 & ,\quad x=0 \end{array} \right.$$ ...
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1answer
23 views

If $f>0$ and $(\ln f)'=f'/f$ is Lipschitz continuous, are we able to conclude that $(\ln f)'''$ is bounded?

Let $f\in C^2(\mathbb R)$ be positive and $g:=\ln f$. Assume $$g'=\frac{f'}f$$ is Lipschitz continuous and hence $g''$ is bounded. Is $g'''$ bounded too?
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2answers
28 views

Lipschitz functions are $o(|x|)$?

Consider a Lipschitz function from $\mathbb{R}\to\mathbb{R}$. Can we say that $\lim_{x\to\infty}\frac{f(x)}{|x|}=0$. Can we also say that $f$ is differentiable. Continuity is quite evident. But linear ...
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1answer
16 views

Understanding step in proof that $|J(v)-J(u)-\langle \nabla J(u),v-u\rangle|\leq \frac{\mu}{2}||v-u||^2$ involviing integral of inner-product.

In this proof that: For $J:\mathbb{E}\to\mathbb{R}$ $\mu$-lipschitz differentiable. Have $\forall u,v \in \mathbb{E}$ $$|J(v)-J(u)-\langle \nabla J(u),v-u\rangle|\leq \frac{\mu}{2}||v-u||^2$$ The ...
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3answers
48 views

Lipschitz continuity implies continuity

Say $\vert f(x)-f(y)\vert \le L\vert x-y\vert$. How to prove the following: $\forall \lim_{n \to \infty} x_n = x_0 \wedge x_0\in \Bbb{R}$: $\lim_{n \to \infty}f(x_n) = f(x_0)$? In other words: how to ...
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1answer
33 views

Is Lipschitz condition on derivatives essentially the same thing as epsilon-delta proof?

I am very confused about Lipchitz gradient/derivative. It's often used in machine learning proofs about gradient descent. Are Lipschitz and epsilon delta proof essentially the same thing? Lipchitz: ...
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1answer
34 views

Unique Solution differential equation system/ linear limited

I have the following differential equation system: $$\begin{align} x'(t)&=\sqrt{1+x^2} +y^3 \sin x -x^7\\ y'(t)&=x(1-y^2 \sin x) \end{align}$$ with $ x(0)=x_0, \ y(0)=y_0$ I have to show, ...
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1answer
41 views

Find a sequence of Lipschitz continuous functions on $[0,1]$ whose uniform limit is $\sqrt{x}$.

Find a sequence of Lipschitz continuous functions on $[0,1]$ whose uniform limit is $\sqrt{x}$, which is a non-Lipschitz function.
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1answer
24 views

Why doesn't separate continuity imply continuity?

Suppose $f: U \rightarrow R$ for some open subset $U$ of $R^2$ is continuous in each variable ie. $f(- , y)$ continuous for each fixed y, and $f(x , -)$ continuous for each fixed x. I know the ...
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0answers
19 views

Composition with Lipschitz map is Lipschitz on Sobolev spaces

Suppose that $F: \mathbb{R}^d \rightarrow \mathbb{R}^d$ is Lipschitz with some constant $L$ and that $F(0)=0$. Then it is clear that $F$ defines a Lipschitz continuous map $L^2(\mathbb{R}^d) \...
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3answers
33 views

Proving a Lipschitz function is continuous

A function $f:D\subset \mathbb R \to \mathbb R$ is lipschitz given that there exists a $L\gt0$ such that $|f(x)-f(y)|\le L|x-y|$ I need to prove this function is then continuous. Is there a best ...
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1answer
18 views

Trying to understand Lipschitz condition and some examples

I'm really new with ODE's and I need your help to understand the Lipschitz function and some examples. First, the theory concepts: A function $f(t, y)$ is said to satisfy a Lipschitz condition in ...
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2answers
25 views

How to show that the natural logarithm is Lipschitz on $[\beta, \infty)$

I want to show the following result: Let $\ln(x)$ have domain $D = [\beta, \infty)$ then $|\ln(x) - \ln(y)| \leq \dfrac{1}{\beta} |x-y|, \forall x,y \in D$ I am confused as to how to prove this ...
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1answer
24 views

Multidimensionally Lipschitz continuous iff Lipschitz continuous in every coordinate

Definition. A function $f$ defined on a set $S \subseteq \mathbb R$ is said to be Lipschitz continuous on $S$ if there exists an $L \geq 0$ such that $$\|f(x_1) - f(x_2)\| \le L\|x_1 - x_2\|$$ for all ...
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37 views

Clarification for a proof about Lipschitz approximation in $W^{1, p}(\mathbb{R}^n)$

I was reading a proof about the Lipschitz approximation of functions $u\in W^{1, p}(\mathbb{R}^n)$. There the author defines a set $$E_{\lambda}=\{x\in\mathbb{R}^n:M|\nabla u|(x)\leq \lambda\}, \quad ...
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1answer
42 views

Show $T(x) := x+sf(x)$ is a bijection with $f$ Lipschitz and $\vert s \vert \lt \frac{1}{L}$

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a function and $L \gt 0$ such that $\Vert f(x)-f(y)\Vert \leq L \Vert x-y\Vert\ $for all $x,y \in \mathbb{R}^n$ Show that $T(x) := x+sf(x)$ defines ...
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1answer
29 views

Is this function involving indicator function Lipschitz?

Is this function $$x1_{\{x>0\}}(x)$$ Lipschitz? It's not differential so mean value cant be used here.
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29 views

Find Lipschtiz constant for a function in matrix

I have the following function in $X \in R^{n \times k}$ $$f(X) = -4A X\Lambda_1 + 4(X\Lambda_1 X^T X\Lambda_1) - 4A^TXY\Lambda_2Y^T + 4XY\Lambda_2Y^TX^TXY\Lambda_2Y^T$$ where $A \in R^{n \times n}$, ...
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0answers
25 views

Lipschitz-Like Condition on Continuously Differentiable, Multivariable Function

If $\epsilon > 0$, $c \ge 0$, $B(x,\epsilon) \subseteq \Bbb{R}^p$, and $\phi : B(x,\epsilon) \to \Bbb{R}^q$ is a continuously differentiable function with $||D\phi(x)|| \le c$ for all $x \in B(x,\...
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0answers
44 views

Picard theorem for $u' = \sqrt{\lvert u^2 -1 \rvert}$ if we know $ u(\pi / 2)= 0$

Problem: can we apply Picard theorem for $$u' = \sqrt{\lvert u^2 -1 \rvert}$$ if $$ u(\pi / 2)= 0$$ [$u$ is a function of a variable $x$ so $u = u(x)$] My attempt: Well, what I need to know is ...
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0answers
22 views

Is the unique solution of autonomous ODE infinitely differentiable

Consider the ODE \begin{align} y'&=f(y)\\ y(0)&=y_0 \end{align} where $f$ is Lipschitz and, if it matters, always positive. When restricted on an interval $[0,a]$, is the unique solution of ...
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2answers
33 views

Show that $|f(x)-L|\leq K|x-c| \implies\lim_{x\to c} f(x)=L$

Let $I$ be an interval in $\mathbb{R}$ such that $f:I\rightarrow\mathbb{R}$ is a function and let $c\in I.$ Show that $|f(x)-L|\leq K|x-c| \implies\lim_{x\to c} f(x)=L$ Proof: Suppose w.l.o.g. that $...
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1answer
27 views

Lipschitz Continuity-Function [closed]

I cannot prove that $A\cos(ax+b)$, where $A$, $a$, $b$ are real numbers is Lipschitz continuous. Am I wrong for trying to do so using the definition?
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0answers
36 views

Lipschitzness of derivatives

Let $f \in C^\infty_b(\mathbb{R}^d; \mathbb{R}^d)$, so bounded, infinitely differentiable with bounded derivatives mapping $\mathbb{R}^d$ to $\mathbb{R}^d$. I'll write $|\cdot|$ for the norm on $\...
3
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1answer
54 views

To verify Lipschitz continuity of the given function $f$.

Consider the function $$f(x,\vec{v}):=g(I_t+\nabla\cdot(I\vec{v}))$$ where $I=I(\vec{x},t)$ is the image intensity function, $g:\mathbb{R}\to[0,\infty)$ is continuous and non negative, $\vec{v}\in H^1(...
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1answer
26 views

Determining whether the function is Lipschitz continuous

I want to determine if the function $f:\mathbb R^2 \to \mathbb R^2$ defined as $f(x,y)=(2x+y,x+2y) $ is Lipschitz continuous. I know that you can consider coordinate functions $f_1,f_2:\mathbb R^2 \...