We define a family of functions $\mathcal{F}$ on a domain $\Omega$ to be equicontinuous if for each point $x \in \Omega$ and any $\epsilon > 0$, there is a $\delta_x > 0$ such that $|f(x) - f(y)| < \epsilon$ whenever $| x - y| < \delta_x$ for all $f \in \mathcal{F}$.
Define a family of functions $\mathcal{F}$ on a domain $\Omega$ to be uniformly equicontinuous if for any $\epsilon > 0$, there is a $\delta > 0$ such that $|f(x) - f(y)| < \epsilon$ whenever $| x - y| < \delta$ for all $f \in \mathcal{F}$ and for all $x \in \Omega$.
The Arzela-Ascoli theorem is sometimes stated for the second class of families, but in fact it is true for both. I am reading a book that proves it only for the second, and they make the comment that the full version is almost never needed in applications. Since this is a complex analysis book I can definitely believe that, but I am wondering if there are any important results that have needed the full strength.
Thanks!