Let $d \colon \mathbb{R} \to \mathbb{R}$. Suppose that I have the following estimate: $$|d(x)-d(y)|\leq C_T |x-y|+e^{-T}$$ for $C_T \to +\infty$ as $T \to \infty$.
Then if I define $$\omega(|x|)=\lim_{T \to \infty} C_T |x|+e^{-T}$$ Then this is a modulus of continuity (i.e. $\omega(0)=0$ and it is subadditive), right?
The if I let $x \to y$ who guarantees that $\omega(|x-y|)\to 0$? I mean this is guaranteed if $T \to \infty$ sufficiently lower than $|x-y|\to0$, right?
But how do you guarantee it in general?