MHB Lipschitz Condition and Uniform Continuity

[Math Amateur]

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:
View attachment 7285In the above text from Bartle and Sherbert we read the following:

"... ... However, there is no number $$K \gt 0$$ such that $$\lvert g(x) \lvert \le K \lvert x \lvert $$ for all $$x \in I$$. ... ... "Can someone please explain why the above quoted statement holds true ...

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

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

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

... so a large $$K$$ is required ... ... and the required number will get larger and larger without bound as $$x$$ gets smaller ...
Is the above the correct explanation for $$f$$ not being Lipschitz on $$I$$ ... ... ?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 ...
View attachment 7286
Note that in the above example B&S take $$u $$ as the point $$u = 0$$ ... ... Peter

 

[GJA]

Hi Peter,

Your reasoning in Edit 1 is correct. Symbolically you could note that for $x\in (0,2]$, $x^{-1/2}\leq K$, verifying your claim that $K$ grows without bound as $x\rightarrow 0^{+}$.