TITLE: Understanding Lipschitz Continuity
QUESTION [20 upvotes]: I have heard of functions being Lipschitz Continuous several times in my classes yet I have never really seemed to understand exactly what this concept really is. 
Here is the definition. 
$\left | f(x_{1})-f(x_{2}) \right |\leq K\left | x_{1}-x_{2} \right |$
Here is the function I'm using. It is known that this is Lipschitz Continuous. 
$f(x)=\sqrt{x^2+5}$
If you pick some points. Here I picked (1, 0.408) and (2, 0.66). 
The result is:
$\left | 0.252 \right |\leq K\left | 1 \right |$
So as long as K is 0.252 or bigger then this function is Lipschitz Continuous?
What if I pick K to be 0.0001 is the function no longer Lipschitz Continuous? 
To me this is hard to understand, why not always pick K to be very large such that the function is always Lipschitz Continuous? 
Unless the left hand side of the inequality is infinity, can't you always find a K big enough to satisfy this inequality?

REPLY [2 votes]: Definition: f is Lipschitz iff there exists $K>0$ such that for all $x,y\in dom(f)$ we have $$(1)\quad |f(x)-f(y)|<K |x-y|.$$ For $S\subset dom(f)$ we say $f$ is Lipschitz on $S$ to mean that $f$ restricted to the domain $S$ is a Lipschitz function. This is most commonly done when $S$ is an interval or a half-line.
The value of $K$ is not unique, for if (1) holds for all $x,y\in  dom (f)$ then it will hold with $K$ replaced by $42K.$ What matters is whether $at$ $least$ $one$ $K>0$ exists such that (1)  holds for all $x,y \in dom(f).$