Recall, given a metric space $X$, a function $f:X \rightarrow \mathbb{R}$ has (uniform) modulus of continuity $w:[0,\infty) \rightarrow [0,\infty]$ if $|f(x) - f(y)| < w(|x-y|)$ for all $x,y \in X$.

In particular, $f$ is $K-$Lipschitz $\implies$ $w(\delta) = K\delta$ works (and in particular, we can choose the 'tightest' $K$ can be chosen). And $f$ is uniformly continuous on $X$ $\implies w(\delta) = \epsilon_{\delta}$, where $\epsilon_{\delta} = \inf\{ \epsilon: |f(x)-f(y)| \leq \epsilon \text{ whenever } |x-y| \leq \delta\}$.

Now, for (uniform) continuity, we also require that $\lim_{\delta \rightarrow 0} w(\delta)=0$. Also, given a uniformly continuous function $f$, we can think about associating it with the best case modulus of continuity, $w_f(\delta) := \inf \{w(\delta): w \text{ is a modulus of cty. for } f \}$.

Now, consider the class of uniformly continuous $f$ with `worse-than-linear' or sublinear $w_f$, such that $\lim_{\delta \rightarrow 0} \frac{w_f(\delta)}{\delta} = \infty $, i.e., as points get closer, the distance between them decreases at a worse-than-linear rate.

Similarly, let $f$ have superlinear modulus of continuity if $\lim_{\delta \rightarrow 0} \frac{w_f(\delta)}{\delta} = 0 $.

Question: Like how the notion of Lipschitz functions = functions with linear modulus of continuity, of which functions with bounded derivatives form a subclass, I'm looking for any sort of alternate characterization (sufficient and/or necessary) of the class of superlinear/sublinear classes.

P.S. Hopefully I'm using sublinear and superlinear in the ways that match usual mathematical intuition!