How do I have an intuitive understanding of modulus of continuity

Question

I briefly chanced upon something called the modulus of continuity while starting on an introductory analysis course on limits and continuity:

$\text{let } f:I \to \mathbb{R}$. Then for all $x,y \in I$, we let the modulus of continuity be $$ \omega =\sup_{x\neq y } \frac{|f(x)-f(y)|}{|x-y|}$$

and if w is finite iff f is uniformly continuous( I was wondering if there's a proof of the converse of this? am I correct to say that seems intuitive because if its not uniformly cont then it means we can find two pairs of points such that no matter how small we shrink the distance between them we can find x,y s.t numerator would be larger than a some constant hence the sup would be unbounded):

The main issue is I am unsure why this fact is useful and can't seem to have an intuitive understanding of why we must define such a modulus of continuity.

Answer 1

If $\omega <\infty$ then $f$ is uniformly continuous but the converse is false. For example if $f(x)=\sqrt x$ in $I=[0,1]$ the $f$ is uniformly continuous but $\omega =\infty$.

Answer 2

Your reading is incorrect. You should understand it intuitively as saying no matter what distinct pairs of points you pick in the interval $I,$ calculate the ratio in question, which is the rate of change in value per change in argument. This is simply the modulus of a difference quotient, the only difference being that both $x$ and $y$ can vary in $I.$ The modulus $\omega$ is then defined as the supremum of this set of absolute difference-quotients. This modulus will exist if the set is bounded above (obviously, it is bounded below by $0,$ but this is irrelevant here), for then a supremum must exist by the completeness of $\mathbf R.$

In short, this quantity just places a bound on how fast a function is changing on average over any subinterval of $I.$