Linear Moduli of Continuity

Let $\mathbb{R}$ be the ordered field of real numbers, with order $\le$, and let $T=\{t\in\mathbb{R}:0\le t\}$. Let $c\in\mathbb{R}$ satisfy $0\le c$, and let $\omega_{c}:T\to\mathbb{R}$ be the function given by

$$\omega_{c}(t)=ct\qquad\text{for }t\in T .$$

Then the following hold.

1. (Modulus of continuity)(i) $\omega_{c}$ is a modulus of continuity.

2. (Monotonicity)(ii) $\omega_{c}(s)\le\omega_{c}(t)$ for all $s,t\in T$ with $s\le t$.