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 \(\displaystyle K \gt 0\) 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 ...
View attachment 7286
Note that in the above example B&S take \(\displaystyle u \) as the point \(\displaystyle u = 0\) ... ... 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 $x\rightarrow 0^{+}$.

 

[math771]

Hi Peter,

You are correct in your understanding of why f is not Lipschitz on I. The statement \lvert g(x) \lvert \le K \lvert x \lvert for all x \in I means that there exists a constant K such that the absolute value of g(x) is always less than or equal to the absolute value of x. However, in this example, as x gets smaller and smaller, the value of \sqrt{x} also decreases and eventually becomes smaller than x, making it impossible to find a constant K that satisfies the condition for all x in I.

In other words, for any given K, we can always find an x in I (specifically, x = \frac{1}{K^2}) where \lvert g(x) \lvert = \frac{1}{K} > \frac{1}{K^2} = \lvert x \lvert, breaking the Lipschitz condition.

I hope this explanation helps clarify the concept. And thank you for including the relevant text from B&S for reference. It is always helpful to have the source material available for better understanding.