I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ...

I am focused on Chapter 5: Continuous Functions ...

I need help in fully understanding an aspect of Example 5.4.6 (b) ...

Example 5.4.6 (b) ... ... reads as follows:

In the above text from Bartle and Sherbert we read the following:

"... ... However, there is no number $ \displaystyle K \gt 0$ such that $ \displaystyle \lvert g(x) \lvert \le K \lvert x \lvert $ for all $ \displaystyle x \in I$. ... ... "

Can someone please explain why the above quoted statement holds true ...

Peter

*** EDIT 1 ***

Just noticed that for $ \displaystyle x$ less than $ \displaystyle 1$ we have $ \displaystyle \sqrt{x}$ is larger than $ \displaystyle x$ ... ...

e.g. $ \displaystyle \sqrt{0.0004}$ is $ \displaystyle 0.02$ ... ... and then we require $ \displaystyle K$ such that ...

$ \displaystyle \lvert 0.02 \lvert \le K \lvert 0.0004 \lvert$

... so a large $ \displaystyle K$ is required ... ... and the required number will get larger and larger without bound as $ \displaystyle x$ gets smaller ...

Is the above the correct explanation for $ \displaystyle f$ not being Lipschitz on $ \displaystyle I$ ... ... ?

Peter

*** EDIT 2 ***

It may be helpful for readers of the above post to have access to B&S's definition of the Lipschitz function/condition ... ... so I am providing the following text from Bartle and Sherbert ...

Note that in the above example B&S take $ \displaystyle u $ as the point $ \displaystyle u = 0$ ... ...

Peter