Sums, Nonnegative Multiples, Monotonicity and Quadratic Reparametrisation of Moduli of Continuity

In the setting of The Real Numbers: Standing Notation and Background, let $T$ be the set of those $t\in\mathbb{R}$ with $0\le t$, let $\omega$ and $\omega'$ be moduli of continuity on $T$, and let $c\in\mathbb{R}$ satisfy $0\le c$. That a function on $T$ with values in $\mathbb{R}$ is nondecreasing is as defined there, with $T$ as the set on which it is considered. Then the following hold.

1. (Sums)(i) The function on $T$ whose value at $t$ is $\omega(t)+\omega'(t)$ is a modulus of continuity.

2. (Nonnegative multiples)(ii) The function on $T$ whose value at $t$ is $c\,\omega(t)$ is a modulus of continuity.

3. (Monotonicity)(iii) If $\omega$ and $\omega'$ are nondecreasing, then so is the function of claim 1; if $\omega$ is nondecreasing, then so is the function of claim 2.

4. (Quadratic reparametrisation)(iv) Let $M\in\mathbb{R}$ satisfy $0\le M$ and assume that $\omega(t)\le M$ for every $t\in T$. For $s\in T$ let

$$Q(s)=\bigl\{\omega(t):\ t\in T,\ t^{2}\le c\,s\bigr\};$$

this set is nonempty, since $0\in T$ and $0^{2}=0\le c\,s$ by claim 5 of Elementary Arithmetic in an Ordered Field, so that it contains $\omega(0)$, and it is bounded above by $M$; hence its supremum $\sup Q(s)$ is defined, as recorded in The Real Numbers: Standing Notation and Background (iv). Let $\omega^{[c]}:T\to\mathbb{R}$ be the function with value $\omega^{[c]}(s)=\sup Q(s)$ at $s\in T$, called the quadratic reparametrisation of $\omega$ at $c$. Then $\omega^{[c]}$ is a nondecreasing modulus of continuity, it satisfies $0\le\omega^{[c]}(s)\le M$ for every $s\in T$, and

$$\omega(t)\le\omega^{[c]}(s)\qquad\text{for all }s,t\in T\text{ with }t^{2}\le c\,s .$$