Purchase Solution

Lipschitz functions

Not what you're looking for?

Ask Custom Question

A function f:A->R is called Lipschitz if there exists a bound M>0 such that Absolute value of f(x)-f(y)/x-y <=M for all x, y belong to A. Geometrically speaking a function f is Lipschitz if there is a uniform bound on the magnitude of the slopes of lines drawn through any two points on the graph of f.

a- Show that if f:A->R is Lipschitz then it is uniformly continuous on A.

b- Is the converse statement true? Are all uniformly continuous functions necessarily Lipschitz?

Purchase this Solution
Solution Summary

This is a proof regarding uniformly continuous and Lipschitz functions.

Solution Preview

a. Proof:
f: A->R is Lipschitz, then there exists M>0, such that for any x,y in A, we have |f(x)-f(y)|<=M|x-y|. Now for any e>0, we can find ...

Purchase this Solution
Free BrainMass Quizzes
Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Probability Quiz

Some questions on probability

View More Free Quizzes
Related BrainMass Solutions

  • Differentiable Functions : Lipschitz and Absolute Values

    108682 Differentiable Functions : Lipschitz and Absolute Values I have a function that is differentiable on [a,b] and I am trying to figure out which scenario is more restrictive: a) the function is a Lipschitz function with a Lipschitz constant L

  • Local or Uniform Lipschitz Constants

    50875 Local or Uniform Lipschitz Constants Determine if the following functions satisfy local or uniform Lipschitz condition. 1). te^y My work: I found d/dy (te^y) = te^y, and this is not bounded above for any value of y, so this made me conclude

  • Lipschitz functions

    Let M_X be the set of all functions f in C_[a,b] satisfying a Lipschitz condition..... The explanations are in the attached text in two formats.

  • Lipschitz Continuity and Initial Value Problems for ODE's

    Find a Lipschitz constant, K, for the function f (u, t) = u^3 + t u^2 which shows that f is Lipschitz in u on the set 0 ? u ? 2, 0 ? t ? 1. PROBLEM 2. Show that the function f (u, t) = t u^(1/2), is not Lipschitz in u on [0, 1] × [0, 2].

  • Proof lipschitz continuous

    This solution helps with proofs for functions.

  • Real analysis : Lipschitz Criterion

    The Lipschitz criterion is investigated.

  • Fredholm equation and Lipschitz condition

    87797 Real Analysis : Fredholm equation and Lipschitz condition Consider the nonlinear Fredholm equation where is continuous on [a,b] and is continuous and satisfies a Lipschitz condition: on the set .

  • Real Analysis : Lipschitz Condition

    152221 Real Analysis using the Lipschitz Condition Please see the attached file for the fully formatted problems. Please see the attached file for the complete solution. A proof using the Lipschitz condition is provided in the solution.

  • Existence and Uniqueness of Solution to an ODE

    C^0 (I)Ã-Lipschitz(L_â?? (R)), f is continuous with respect to 1st variable and Lipschitz with respect to 2nd variable. Lipschitz Continuity: A function g:Iâ?'R is Lipschitz continuous if â??Î?>0 such that â?-g(â-x)-g(â-y)â?-â?¤Î?â?

View More Related Solutions