Definition. equicontinuous [br722]

A family $\mathscr{F}$ of complex functions $f$ defined on a set $E$ in a metric space $X$ is said to be equicontinuous on $E$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that $$ \lvert f(x) - f(y) \rvert < \epsilon $$ whenever $d(x, y) < \delta, x \in E, y \in E$, and $f \in \mathscr{F}$. Here $d$ denotes the metric of $X$.

It is clear that every member of an equicontinuous family is uniformly continuous.