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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

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

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

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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