Modulus of Continuity

Let $\mathbb{R}$ be the ordered field of real numbers and let $T=\{t\in\mathbb{R}:0\le t\}$.

A function $\omega:T\to\mathbb{R}$ is a modulus of continuity if the following two conditions hold.

1. $0\le\omega(t)$ for every $t\in T$.

2. For every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there exists $\delta\in\mathbb{R}$ with $0<\delta$ such that every $t\in T$ with $t\le\delta$ satisfies