Lipschitz Definition and 69 Threads

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...

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...

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...

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...

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...

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...

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...

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?

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...

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...

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?

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...

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^*|| <=...

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...

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...

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...

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' \\...

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...

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...