Example of $M \subset \mathcal{C}_0 $ where $\lim_{\delta \to 0} \sup_{f \in M} w_{\delta}(f) > 0$

Question

Let $\mathcal{C}_0 := \{f: [0,1] \to \mathbb{R} \text{ continuous}, f(0) = 0\}$. I'm looking for an example of a subset $M \subset \mathcal{C}_0 $ with

$$ \lim_{\delta \to 0} \sup_{f \in M} w_{\delta}(f) > 0 $$

where

$$w_{\delta}(f) := \sup \{|f(s) - f(t)|: s,t \in [0,1], |s-t| \le \delta\}$$ It holds that $\forall f \in \mathcal{C}_0: \lim_{\delta \to 0}w_{\delta}(f) = 0$.

Answer 1

Let $f_n(x)=x^n$. For $\delta \in (0, 1)$, notice that $$ \omega _\delta (f_n) \geq f(1) - f(1-\delta ) = 1-(1-\delta )^n. $$

Setting $M=\{f_n:n\in {\mathbb N}\}$, it follows that $$ \sup_{f \in M} w_{\delta}(f) \geq \sup_{n\in {\mathbb N}} 1-(1-\delta )^n = 1, $$ and therefore $$ \lim_{\delta \to 0} \sup_{f \in M} w_{\delta}(f) = 1. $$