I am currently in a course which is going through rigorous definitions of continuity, convergence, integrals, etc. I am trying to develop an intuition in understanding the modulus of continuity, defined for a function $f : S \to \mathbb{R}$ as
$\omega_f (\delta) = \text{l.u.b.}|f(s^\prime) - f(s)| \quad \forall s, s' \in S : |s' - s| < \delta$.
So how I understand it is that, given all pairs of points $s^\prime$ and $s$ such that they are within a distance of $\delta$ of each other, the largest difference between the functions evaluated at those points is the value of $\omega_f$. However, this is still a bit unsatisfying since it's a bit of an abstract idea. Is there a better way to understand this?
The definition of the modulus of continuity $\omega$ I'm using is
$$ \omega(f,\delta) := \max\{ |f(x) - f(y)|: x\in S, |x-y|\le \delta\},$$
which I interpreted as a window of size $\delta$ skimming through $S$ to find the interval containing $(x^*, y^*)$ which map on the two values $f(x^*)$ and $f(y^*)$ that are farest away from each other.