This is actually stronger than the epsilon-delta definition of continuity, as noted in the wiki article -- "For any α > 0, the condition implies the function is uniformly continuous."

Normally, you'd have something like

For all ϵ > 0, there exists δ(x, ϵ) > 0 such that

if |x - x₀| < δ, then |f(x) - f(x₀)| < ϵ.

(namely, at any given point, there's a ball of nearby points such that all their corresponding outputs are close together.)

But, with absolute continuity, δ is purely a function of ϵ, namely, that ball doesn't change size even as we move further away.

For instance, f(x)=x is absolutely continuous, while f(x)=x^2 is not. (for the former case, we can use δ = ϵ/2; for the latter case, we'd need a δ that looks something like 2ϵ·x)

In this case, since the Holder condition depends only on the distance between x and y, it automatically implies absolute continuity.