According to Wikipedia, the modulus of continuity is "used to measure quantitatively the uniform continuity of functions", and is defined as follows:For a function $f: I \rightarrow \mathbf{R}$ admits $\omega$ as a modulus of continuity if
$$|f(x)-f(y)| \leq \omega(|x-y|)$$
for all $x$ and $y$ in the domain of $f$, such that $\omega (t) \to 0$ as $t \to 0$.
A function is uniformly continuous if and only if it admits a modulus of continuity
In Taylor's PDEs I in page 133 the following is stated
Let $\omega: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$be a modulus of continuity, i.e., $\omega(0)=0, \omega$ is continuous, and increasing. We may as well assume $\omega$ is bounded and $C^{\infty}$ on $(0, \infty)$.
My question is, on what basis we can assume $\omega$ is bounded and $C^{\infty}$ on $(0, \infty)$?
Can we expect every uniformly continuous function to have a bounded $\omega$?
Alternatively, $\omega$ is sometimes defined by$$\omega_f(\delta) = \sup_{|x-y| \leq \delta} |f(x) - f(y)|.$$
From this definition, it is obvious that if $f$ is bounded then $\omega_f$ is also bounded.
Is this also true?