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I am focused on Chapter 1: Continuity ... ...

I need help with an aspect of Example 1.8.14 ... ...

The start of Duistermaat and Kolk's Example 1.8.14 reads as follows:https://www.physicsforums.com/attachments/7753In the above example we read the following:

"... ... any linear mapping \(\displaystyle A \ : \ \mathbb{R}^n \rightarrow \mathbb{R}^p\), whether bijective or not, is Lipschitz continuous and therefore uniformly continuous ... ... "

**I am somewhat unsure regarding proving this statement ... but I think the proof goes like the following:**

For \(\displaystyle A \ : \ \mathbb{R}^n \rightarrow \mathbb{R}^p\) to be Lipschitz continuous we require

\(\displaystyle \mid \mid Ax - Ax' \mid \mid \le k \mid \mid x - x' \mid \mid \text{ where } x, x' \in \text{dom} (A)\) ... ... ... (1)now we have:

\(\displaystyle \mid \mid Ax \mid \mid \le k \mid \mid x \mid \mid\) ... ... ... (2)Now ... maybe put \(\displaystyle x = x' - x''\)

... then (2) becomes \(\displaystyle \mid \mid A(x' - x'') \mid \mid = \mid \mid Ax' - Ax'' \mid \mid \le k \mid \mid x' - x'' \mid \mid\)

which is the required result ...

**Is that correct?**Peter