Let \(S\subseteq\CC\) and let \((f_n)_{n\ge1}\) be a sequence of functions \(f_n\colon S\to\CC\). The family \(\{f_n\}\) is equicontinuous on \(S\) if for every \(\varepsilon>0\) there exists \(\delta>0\) such that \[ \abs{f_n(x)-f_n(y)}<\varepsilon\qquad\text{for all } n\ge1 \text{ and all } x,y\in S \text{ with } \abs{x-y}<\delta. \]
The number \(\delta\) depends only on \(\varepsilon\), not on \(n\), \(x\), or \(y\). [@Rud76] states this definition for families of complex functions on a subset of a metric space.
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