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. docnet

    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. Euge

    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. P

    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. hilbert2

    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. T

    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. mertcan

    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. Math Amateur

    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. Math Amateur

    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. Math Amateur

    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. Math Amateur

    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. T

    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. JasMath33

    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. KF33

    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. S

    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...
  15. evinda

    MHB Lipschitz Condition: Does $f(t,y)$ Satisfy? Find Constant

    Hello! (Wave) Does the following $f(t,y)$ satisfy the Lipschitz condition as for $y$, uniformly as for $t$? If so, find the Lipschitz constant. $$f(t,y)=\frac{|y|}{t}, t \in [-1,1]$$ I have tried the following: $$\frac{|f(t,y_1)-f(t,y_2)|}{|y_1-y_2|}=\frac{|y_1|-|y_2|}{t|y_1-y_2|} \leq -...
  16. evinda

    MHB Local Lipschitz condition

    Hello! (Wave) The local Lipschitz condition is the following:Let $c>0$ and $f \in C([a,b] \times [y_0-c, y_0+c])$. If $f$ satisfies in $[a,b] \times [y_0-c,y_0+c]$ the Lipschitz criterion as for $y$, uniformly as for $t$, $$\exists L \geq 0: \forall t \in [a,b] \ \forall y_1, y_2 \in...
  17. P

    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...
  18. M

    MHB Show that it is Lipschitz

    Hey! :o Let $E \subset \mathbb{R}^d$ Lebesgue measurable and $\phi (t)=m \left ( \Pi_{i=1}^{d} (-\infty , t_i ) \cap E \right )$. To show that $\phi$ is Lipschitz, can we do it as followed?? Let $x>y$. $$|\phi(x) - \phi(y)|=|m \left ( \Pi_{i=1}^{d} (-\infty , x_i ) \cap E \right )-m \left (...
  19. S

    MHB Equilibrium points and lipschitz functions

    Given a function Hn-Hn is localy lip on its domain and f(p)=0 for some p an element of D if x is a solution of the system x' =f(x) on interval I and x(s) not equal to p for some s in I. how can i show that x(t) is not equal to p for all t in I
  20. J

    Find region for which F(x,y) = (x+y)^2 is Lipschitz in y

    As the title says, I need to find such a region. Taking any x, and any y1 and y2 I used the expression |F(x,y1) - F(x,y2)| and plugged in the function respectively for y1 and y2. Now I have to find values for x and y such that the following condition (Lipschitz condition) is satisfied: | 2x +...
  21. O

    MHB Proving Local Lipschitz Property for Linear Functions on Real Numbers

    how do i prove that f= mx+c has a local lipschitz property on R
  22. S

    MHB Working on lipschitz function and contraction map

    if you given a function f from R^2 to R^2 f(x)=<f_1(x),f_2(x)>, x in R^2 with f_1 and f_2 from R^2 to R being differentiable on R. if there is contants K_1 and K_2 greater than or equal to 0 so the 2-norm of (gradient f_1(x)) is less than or equal to K_1 and 2-norm of (gradient f_2(x)) is...
  23. O

    Prove that Locally Lipschitz on a Compact Set implies Lipschitz

    Homework Statement Let M and N be two metric spaces. Let f:M \to N. Prove that a function that is locally Lipschitz on a compact subset W of a metric space M is Lipschitz on W. A similar question was asked here...
  24. evinda

    MHB Is $f(x)=\sqrt{x},x\geq 0$ Lipschitz at $[0,\infty]$?

    Hi! :rolleyes: I have also an other question... Could you explain me why $f(x)=\sqrt{x},x\geq 0$ is not Lipschitz at $[0,\infty]$??How can I show this??Do I have to use the condition $|f(x)-f(y)| \leq M|x-y|,M>0$ ,to show this??
  25. M

    Lipschitz function in Real Analysis

    Homework Statement Let f be a real function defined on the interval [a,b]/0<a<b:\forall x,y\in[a,b],x\neq y/|f(x)-f(y)|<k|x^{3}-y^{3}| where k is a positive real constant. Homework Equations 1- Prove that f is uniformly continuous on [a,b] 2- We define a function g on [a,b] such that...
  26. M

    Locally Lipschitz function implications

    Homework Statement . Let ##f:\mathbb R \to \mathbb R##, ##x_0, α \in \mathbb R##. ##f## is locally Lipschitzof of order ##α## at the point ##x_0## if there are ##ε, M>0## such that ##|f(x)-f(x_0)|<M|x-x_0|^α## for every ##x :0< |x-x_0|<ε## Prove that: 1)If ##f## is locally Lipschitz of order...
  27. K

    MHB Proving that a system of twelve ODEs satisfies Lipschitz condition

    I need to know how I can prove the existence and uniqueness of a solution (using Lipschitz condition and well-posedness, stability analysis, etc.) for a system of 12 ordinary differential equations. I have the theorem that I need to use, but the number of calculations and work that I would have...
  28. C

    Using Lipschitz continuity on open intervals

    Homework Statement Prove whether f(x) = x^3 is uniformly continuous on [-1,2) Homework Equations The Attempt at a Solution I used Lipschitz continuity. f has a bounded derivative on that interval, thus it implies f is uniformly continuous on that interval. But as it is not a...
  29. M

    Space of lipschitz functions: two metrics are topologically equivalent

    Homework Statement . Let ##Lip_{M}(ℝ)##={##f: [0,1]→ℝ : |f(x)-f(y)|≤M|x-y|##}. Prove that ##(Lip_{M}(ℝ),d_∞)## and ##(Lip_{M}(ℝ),d_1)## are topologically equivalent but not equivalent. ##d_∞(f,g)=Sup_{x \in [0,1]}|f(x)-g(x)|## ##d_1(f,g)=\int_0^1 |f(x)-g(x)|dx## The attempt at a solution. I am...
  30. F

    Continuity ##f:\mathbb_{R}^3 \to \mathbb_{R}## with Lipschitz

    Homework Statement Prove ## f(x,y,z)=xyw## is continuos using the Lipschitz condition Homework Equations the Lipschitz condition states: ##|f(x,y,z)-f(x_0,y_0,z_0)| \leq C ||(x,y,z)-(x_0,y_0,z_0)||## with ##0 \leq C## The Attempt at a Solution...
  31. G

    Lipschitz vs uniform continuity.

    What is the difference between Lipschitz continuous and uniformly continuous? I know there different definitions but what different properties of a function make them one or the other(or both). So Lipschitz continuity means the functions derivative(gradient) is bounded by some real number and...
  32. S

    Numerical theory and Lipschitz function

    so I have this homework as I said and marks will be added on my total, so if anyone could help you will be a lifesaver, you don't have to answer the whole thing , just help me with the part you know, here it is : A function g (x) is called Lipschitz function on the interval [a,b] if there...
  33. marellasunny

    When does a Lipschitz condition fail for a DE?

    could you give an example where the Lipschitz condition fails,like when there is a periodic forcing function? I'm thinking the Lipschitz condition would fail for a non-autonomous differential system because period-2 orbits exist for 2D non-autonomous continuous dynamical systems,which means the...
  34. B

    Lipschitz Condition, Uniqueness and Existence of ODE

    Homework Statement Find a solution of the IVP \frac{dy}{dt} = t(1-y2)\frac{1}{2} and y(0)=0 (*) other than y(t) = 1. Does this violate the uniqueness part of the Existence/Uniqueness Theorem. Explain. Homework Equations Initial Value Problem \frac{dy}{dt}=f(t,y) y(t0)=y0 has a...
  35. B

    Lie derivatives and Lipschitz contants

    Hello all, I have a problem with an inequality. Let Is the following proof valid? from which, taking the norm to both sides yields where L is the Lipschitz constant of f w.r.t. x. Thus, can I conclude that Is it correct? Thanks :)
  36. F

    Derivatives and continuity / Lipschitz equation

    Hi! I think I've managed to solve this problem, but I'd like it to be checked Homework Statement show that if $$f : A\subset \mathbb{R}\to \mathbb{R}$$ and has both right derivative: $$f_{+}'(x_0),$$ and left derivative $$f_{-}'(x_0)$$ in $$x_0\in A$$, then $$f$$ is continuos in $$x_0.$$...
  37. L

    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. B

    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. L

    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. O

    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. J

    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. A

    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. I

    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. C

    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. I

    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. L

    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. K

    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. C

    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. C

    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. T

    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...