Lipschitz Continuity .... and Continuity in R^n ....

In summary, the conversation focuses on the definitions of continuity and Lipschitz continuity, with further discussion on how to prove that Lipschitz continuity implies continuity and uniform continuity. It is shown that for a given epsilon, choosing delta as epsilon over the Lipschitz constant k works for all x, thus proving uniform continuity.
  • #1
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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 definitions ...)I have tried to show that Lipschitz Continuity implies continuity of a mapping f ... but have not succeeded ...

Can someone please demonstrate how to rigorously prove that Lipschitz continuity implies that f is continuous ... (***edit*** ... better still would be to show that Lipschitz continuity implies that f is uniformly continuous ...)Help will be appreciated ...

Peter========================================================================================***Note***

D&K's definitions of continuity and Lipschitz continuity read as follows:https://www.physicsforums.com/attachments/7749Peter
 
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  • #2
Hi, Peter.

For a given epsilon, choose $\delta=\epsilon/k$. Note that the choice of $\delta$ works for all $x$, which implies uniform continuity.
 
  • #3
GJA said:
Hi, Peter.

For a given epsilon, choose $\delta=\epsilon/k$. Note that the choice of $\delta$ works for all $x$, which implies uniform continuity.

Thanks GJA ... ...

I think I can see this ... since then we have ...

\(\displaystyle \mid \mid x - x' \mid \mid \Longrightarrow \mid \mid f(x) - f(x') \mid \mid\le k \mid \mid x - x' \mid \mid = k \delta = \epsilon
\)Is that correct?

Peter
 
  • #4
You got it!
 

Related to Lipschitz Continuity .... and Continuity in R^n ....

1. What is Lipschitz continuity?

Lipschitz continuity is a mathematical concept that describes the behavior of a function. A function is considered Lipschitz continuous if there exists a constant value, called the Lipschitz constant, such that the distance between the function values at any two points is always less than or equal to the product of the Lipschitz constant and the distance between the two points.

2. How is Lipschitz continuity different from regular continuity?

The main difference between Lipschitz continuity and regular continuity is the use of a constant value in the definition. While regular continuity only requires the function values to approach each other as the points get closer, Lipschitz continuity puts a limit on how fast the values can change. This can be seen as a stronger condition for continuity.

3. How is Lipschitz continuity related to differentiability?

Lipschitz continuity implies differentiability, but the reverse is not always true. If a function is Lipschitz continuous, it must also be differentiable at every point where it is defined. However, there are functions that are differentiable but not Lipschitz continuous, such as the absolute value function.

4. What is continuity in R^n?

Continuity in R^n refers to the behavior of a function with multiple variables. In this case, the function must satisfy the condition that as the points get closer in n-dimensional space, the function values also get closer. This is similar to the definition of continuity in one dimension, but it applies to multiple dimensions.

5. How is continuity in R^n related to topology?

Topology is the branch of mathematics that deals with the properties of objects that are unchanged by continuous deformations. Continuity in R^n is closely related to topology because it defines a function that is continuous in all dimensions, which is a key concept in topology. In fact, continuity in R^n is often used to define topological spaces in higher dimensions.

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