# Thread: Lipschitz Condition and Uniform Continuity

1. 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

2. 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 $x\rightarrow 0^{+}$.

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