# What is Lipschitz: Definition and 69 Discussions

Lipschitz, Lipshitz, or Lipchitz is an Ashkenazi Jewish surname. The surname has many variants, including: Lifshitz (Lifschitz), Lifshits, Lifshuts, Lefschetz; Lipschitz, Lipshitz, Lipshits, Lopshits, Lipschutz (Lipschütz), Lipshutz, Lüpschütz; Libschitz; Livshits; Lifszyc, Lipszyc. It is commonly Anglicized as Lipton, and less commonly as Lipington.
There are several places in Europe from where the name may be derived. In all cases, Lip or Lib is derived from the Slavic root lipa (linden tree, see also Leipzig), and the itz ending is the Germanisation of the Slavic place name ending ice.
In the Czech Republic:

Libčice nad Vltavou (German: Libschitz an der Moldau)
Liběšice u Litoměřic (German: Liebeschitz bei Leitmeritz)
Liběšice u Žatce (German: Libeschitz bei Saaz)In Poland:

Głubczyce (Silesian German: Lischwitz, German: Leobschütz)In mathematics, the name can be used to describe a function that satisfies the Lipschitz condition, a strong form of continuity, named after Rudolf Lipschitz.
The surname may refer to:

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1. ### Assume that if the real-valued function h(x) is Lipschitz continuous...

I think the answer is no, since the requirements for Lipschitz continuous and epsilon-delta continuous are different. The reason I'm asking such an odd question is, I made a mistake by writing a proof of the Lipschitz continuity of ##g(h(x))## using the assumption that ##h(x)## is Lipschitz...
2. ### POTW A Series Converging to a Lipschitz Function

Prove that the series $$\sum_{k = 1}^\infty \frac{(-1)^{k-1}}{|x| + k}$$ converges for all ##x\in \mathbb{R}## to a Lipschitz function on ##\mathbb{R}##.
3. ### Find domain where function is Lipschitz

The reduction is simple in all cases. For the first one, put ##x_1=x, x_2=x'## and ##x_3=x''##. Let ##\pmb{x}=(x_1,x_2,x_3)##. Then we get $$\pmb{x}'= \begin{pmatrix}x_1' \\ x_2' \\ x_3' \end{pmatrix}=\begin{pmatrix}x_2 \\ x_3 \\ 1-x_1^2 \end{pmatrix}=\pmb{f}(\pmb{x}),$$ where...
4. ### I Hölder and log-Hölder continuity

Now, there's this conventional definition of the Hölder continuity of a function ##f## defined on ##[a,b]\subset\mathbb{R}##: For some real numbers ##C>0## and ##\alpha >0##, and any ##x,y\in [a,b]##, ##|f(x) - f(y)|<C|x-y|^{\alpha}##. However, this does not include functions like ##f(x) =...
5. ### Is f(t,y)=e^{-t}y Lipschitz Continuous in y?

This is not so much a "Homework" question I am just giving an example to ask about a specific topic. Homework Statement Is ##f(t,y)=e^{-t}y## Lipschitz continuous in ##y## Homework Equations I don't really know what to put here. Here is the definitions...
6. ### I Lipschitz Condition: Finding the Lipschitz Constant

Hi, as I see Lipschitz condition is written as: |f(x)-f(x')| <= M*|x-x'| and minimum M is called Lipschitz constant. I would like to ask how the minimum M is found out? For instance for many convergence theorem include Lipschitz condition and no say something about value of M but how M is...
7. ### MHB Linear Mappings are Lipschitz Continuous .... D&K Example 1.8.14 .... ....

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Example 1.8.14 ... ... The start of Duistermaat and Kolk's Example 1.8.14 reads as...
8. ### MHB The Euclidean Norm is Lipschitz Continuous .... D&K Example 1.3.5 .... ....

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Example 1.3.5 ... ... The start of Duistermaat and Kolk's Example 1.3.5 reads as...
9. ### MHB Lipschitz Continuity .... and Continuity in R^n ....

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... In Definition 1.3.4 D&K define continuity and then go on to define Lipschitz Continuity in Example 1.3.5 ... ... (see below for these...
10. ### MHB Lipschitz Condition and Uniform Continuity

I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 5: Continuous Functions ... I need help in fully understanding an aspect of Example 5.4.6 (b) ...Example 5.4.6 (b) ... ... reads as follows: In the above text...
11. ### I Finding a Lipschitz Constant

Compute a Lipschitz constant K as in (3.7) $$f(t, y_2)-f(t, y_1)=K(y_2-y_1) \space\space (3.7)$$, and then show that the function f satisfies the Lipschitz condition in the region indicated: $$f(t, y)=p(t)\cos{y}+q(t)\sin{y},\space {(t, y) | \space |t|\leq 100, |y|<\infty}$$ where p,q are...
12. ### Lipschitz Q: Show w/ Example & Derivative

Homework Statement Homework EquationsThe Attempt at a Solution I know I will just have to show this by one example. I thought about using f(x) = x2 but I am not sure if this satisfies the last part dealing with the absolute value of the derivative. It is just the last part on which I am stuck.
13. ### I What is the relationship between contractive and Lipschitz functions?

I was looking at this definition of a contractive function and the only difference I saw between it and a Lipschitz function was the b and M. I am just wondering how you look at the connections between them.
14. ### Lipschitz perturbations and Hammerstein integral equations

Recently I was a witness and a minor contributor to this thread, which more or less derailed, in spite of the efforts by @Samy_A. This is a pity and it angered me a bit, because the topic touches upon some interesting questions in elementary functional analysis. Here I would like to briefly...

17. ### Lipschitz condition and Leibniz rule

Hi, I am reading a paper that states: "We note that if an integrable function satisfies the Lipschitz condition of order one, then differentiation and integration can be interchanged. This provides a more compact way to take the derivative. Consequently, in our proofs, if an integrable function...

37. ### Lipschitz Continuous: Check Solutions & Get Hints

This question is about lipschitz continuous, i think the way to check if the solutions can be found as fixed points is just differentiating f(t), but I'm not sure about this. Can anyone give me some hints please? I will really appreciate if you can give me some small hints.
38. ### Help interpreting HW question on Lipschitz Hölder

Homework Statement I only need help interpreting the following: Show that every Lipschitz continuous function is α-Hölder continuous for every α ∈ (0, 1 The definition of both is given in the homework so this seems trivial but it's a graduate level class. Am I mising something? Thanks...
39. ### Global and local Lipschitz proof

Homework Statement f(x)={0 for x<0, \sqrt{x} else} a) Is f(x) globally Lipschitz? Explain b) Find the area for which f(x) is locally Lipschitz. Homework Equations The Attempt at a Solution a) f(x) is not globally Lipschitz in x on [a,b]xRn since there is a discontinuity at x=0. b) I would...
40. ### Lipschitz Property of Norms: Comparing α-norm and β-norm in ℝn

Homework Statement Hello friends, i couldn't find a solution for the question below. Can you help me? Thank you very much. Let α-norm and β-norm be two different norms on ℝn. Show that f:ℝn->ℝm is Lipschitz in α-norm if and only if it is Lipschitz in β-norm Homework Equations...
41. ### Hint needed for lipschitz problem.

a sufficient condition for uniqueness is the Lipschitz condition: ￼ On a domain D of the plane, the function f (x, y) is said to satisfy the Lipschitz condition for a constant k > 0 if: |f(x,y1)−f(x,y2)|≤k|y1−y2| for all points (x,y1) and (x,y2) in D. Give an example of an IVP with...
42. ### Integrability and Lipschitz continuity

(I've been lighting this board up recently; sorry about that. I've been thinking about a lot of things, and my professors all generally have better things to do or are out of town.) Is there an easy way to show that if f is Lipschitz (on all of \mathbb R), then \int_{-\infty}^\infty f^2(x)...
43. ### Lipschitz function and Baire Category Theorem

hey, I need to show, using Baire Category Theorem, that there exits a continuous function f: [0,1] to R , that isn't Lipschitz on the interval [r,s] for every 0<=r<s<=1 . I defined the set A(r,s) to be all the continuous functions that are lipschitz on the interval [r,s]. I showed that...
44. ### Lipschitz condition and blow up time

Hello everyone, I have asked a similar question in the DE forum but couldn't get an answer so I'm hoping the mods will be tolerant and let me post it here even though it's not strictly analysis. I'm considering a DE of the form x' = f(t, x) where f is a continuous function defined on an open...
45. ### Proving Real Lipschitz Function Differentiability

Hello I've been told that a (real) Lipschitz function (|f(x)-f(y)|<M|x-y|, for all x and y) must be differentiable almost everywhere. but I don't see how I can prove it. anyone has an idea? Thanks
46. ### Lipschitz Continuous Functions: Differentiable Almost Everywhere?

Someone told me that a continuous function that is Lipschitz is differentiable almost every where. If this is true how do I see it?
47. ### F(x)=||x|| is Lipschitz function

"Let (V,||.||) be a normed vector space. Then by the triangle inequality, the function f(x)=||x|| is a Lipschitz function from V into [0,∞)." I don't understand how we this follows from the triangle inequality. How does the proof look like? Any help is appreciated!
48. ### Function is lipschitz continuous

Homework Statement prove that if f is continuously differentiable on a closed interval E, then f is Lipschitz continuous on E. The Attempt at a Solution so I'm letting E be [a,b] I'm using the mean value theorem to show secant from a->b = some value, then I'm saying if I subtract...
49. ### Lipschitz Continuity Proof: f(x) = x^(1/3) on (-1,1) Has No Lipschitz Constant

Homework Statement Show f(x) = x^(1/3) is not lipschitz continuous on (-1,1). Homework Equations I have abs(f(x)-f(y)) <= k*abs(x-y) when I try to show that there is no K to satisfy I have problems
50. ### Show Lipschitz and Uniform Continuity of f(x)=xp on [a,b]

Let f(x)=xp Show that f is Lipschitz on every closed sub-interval [a,b] of (0,inf). For which values of p is f uniformly continuous. So, we know that the map f is said to be Lipschitz iff there is a constant M s.t. |f(p)-f(q)|<=M|p-q|. And we were given the hint to use the Mean Value...