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

[Math Amateur]

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

 

[GJA]

Hi, Peter.

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

 

[Math Amateur]

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 ...

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

Peter

 

[GJA]

You got it!