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. C

    Lipschitz functions dense in C0M

    I'm working on Pugh's book on analysis and there's this problem that should be very easy to solve. It's asking to show that the set of continuous functions, f:M \rightarrow R, f\in C^{Lip} obeying the Lipschitz condition (where M is a compact metric space): |f(a) - f(b)| \leq L d(a,b) for...
  2. Z

    Is the Function |x| Locally Lipschitz?

    Hi, I am struggling with the concept of "locally Lipschitz". I have read the formal definition but i cannot see how that differs from saying something like: "A function is locally Lipschitz in x on domain D if the function doesn't blow up anywhere on D"? It seems that, when talking about...
  3. W

    Is the Derivative of a Multivariable Function Lipschitz?

    Here is a tough one: Say we have a multivariable function f:R^n -> R and for any x, and direction u, the function g:R->R defined as g(t)=f(x+tu) has that g'(t) is Lipschitz with the same Lipschitz constant (say M). For special cases, taking u to be any basis element we see that every partial...
  4. quasar987

    Question about Lipschitz continuity

    Homework Statement This should be easy but I'm stomped. Let K be a compact set in a normed linear space X and let f:X-->X be locally Lipschitz continuous on X. Show that there is an open set U containing K on which f is Lipschitz continuous.Homework Equations locally Lipschitz means that for...
  5. C

    Proving Set of Lipschitz Functions of Order b in Order a

    Homework Statement 1. Let 0 < a < b <= 1. Prove that the set of all Lipschitz functions of order b is contained in the set of all Lipschitz functions of order a. 2. Is the set of all Lipschitz functions of order b a closed subspace of those of order a? Homework Equations I know...
  6. D

    Sequences of Lipschitz Functions

    Let )<C<\infty and a,b \in \mathbb{R}. Also let Lip_{C}\left(\left[a,b\right]\right) := \left\{f:\left[a,b\right]\rightarrow \mathbb{R} | \left|f(x) - f(y)\right| \leq C \left|x-y\right| \forall x,y \in \left[a,b\right]\right\} . Let \left(f_{n}\right) _{n \in \mathbb(N)} be a sequence of...
  7. V

    Negative Lipschitz (Hölder) exponent: Intuition

    Negative Lipschitz (Hölder) exponent: Intuition! Hi everybody! Sorry for double posting... :( I have some problem with Hölder (Lipschitz) exponent! From what I know Lipschitz refer to integer values whereas Hölder to non integer ones. The usual definition roughly states that: A...
  8. M

    Lipschitz Condition and Differentiability

    Let K>0 and a>0. The function f is said to satisfy the Lipschitz condition if |f(x)-f(y)|<= K |x-y|a .. I am given a problem where I must prove that f is differentiability if a>1. I know I need to show that limx->c(f(x)-f(c))/ (x-c) exists. I am having quite a hard time. Any hints?
  9. R

    Uniform convergence of Fourier Series satisfying Lipschitz condition

    Homework Statement f is integrable on the circle and satisfies the Lipschitz condition (Holder condition with a=1). Show that the series converges absolutely (and thus uniformly). i literally spent about 20 hours on this problem today but i just could not figure it out. i have a feeling...
  10. A

    Lipschitz function and uniform continuity

    A function f:D\rightarrowR is called a Lipschitz function if there is some nonnegative number C such that absolute value(f(u)-f(v)) is less than or equal to C*absolute value(u-v) for all points u and v in D. Prove that if f:D\rightarrowR is a Lipschitz function, then it is uniformly...
  11. G

    Locally Lipschitz trig function

    Would a trig function like tan \left(x\right) be locally Lipschitz? How do we know that, if we know that tan \left(x\right) is not continuously differentiable?
  12. W

    Lipschitz ODE Problem: Proving Inequality for Locally Lipschitz Function

    Homework Statement Suppose the function f(t,x) is locally Lipschitz on the domain G in R^2, that is, |f(t,x_1)-f(t,x_2)| <= k(t) |x_1 - x_2| for all (t, x_1),(t,x_2) in G. Define I = (a,b) and phi_1(t) and phi_2(t) are 2 continuous functions on I. Assume that, if (t, phi_i(t)) is in G, then the...
  13. L

    Solving equation numerical using lipschitz

    Homework Statement This problem is about solving an equation system numerical using lipschitz method or whatever the name is. x_1 = sqrt(1-x^2) x_2 = sqrt((9-5x_1^2)/21) and we create (x_1,x_2) = x g(x) = (x_1,x_2) (fixed point equation) lipschitz states: ||x^(k+1) - x^*|| <=...
  14. B

    Lipschitz Continuity and measure theory

    Hi, this is not a homework problem because as you can see, all schools are closed for the winter break. But I'm currently working on a problem and I'm not sure how to begin to attack it. Here's the entire problem: Let f be bounded and measurable function on [0,00). For x greater than or...
  15. B

    Lipschitz Continuity & Uniform Continuity: Showing sinx & cosx in R

    Homework Statement Show that Lipschitz continutity imples uniform continuity. In particular show that functions sinx and cosx are uniformly continuous in R. The Attempt at a Solution I said that if delta=epsilon/k that Lipschitz continuity imples continuity. Now I am stuck as to how to...
  16. R

    Lipschitz condition, more of like a clarification

    Hello, I just have one question that's been bothering me. When I reduce a higher ODE to a First ODE, and if I prove that First ODE satisfies the Lipschitz condition, does that mean that the higher ODE has a unique solution (thanks to some other theorem)? All clarifications are appreciated...
  17. I

    Proving Lipschitz Condition for F with $\| \cdot \|_{1}$ Norm

    I need to prove that the function F is Lipschitz, using the \| \cdot \|_{1} norm. that is, for t \in \mathbb{R} and y, z \in Y(t) \in \mathbb{R}^{2} I must show that \|F(t, y) - F(t, z)\|_{1} < k|y-z| F(t, Y(t)) is given as F(t, Y(t)) = \left( \begin{array}{cc} y' \\...
  18. A

    Lipschitz continuity problem

    Let the function f:[0,\infty) \rightarrow \mathbb{R} be lipschitz continuous with lipschits constant K. Show that over small intervalls [a,b] \subset [0,\infty) the graph has to lie betwen two straight lines with the slopes k and -k. This is how I have started: Definition of lipschits...
  19. K

    Lipschitz Continuity and Uniqueness

    Dear all, If a differential equation is Lipschitz continuous, then the solution is unique. But what about the implication in the other direction? I know that uniqueness does not imply Lipschitz continuity. But is there a counterexample? A differential equation that is not L-continuous, still...
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