Equicontinuity

A family $\mathcal{F}$ of maps between metric spaces is equicontinuous when, near each point, one input tolerance controls the output variation uniformly for every $f\in \mathcal{F}$. Uniform equicontinuity uses one input tolerance over the whole domain.

A family $\mathcal{F}$ of maps between metric spaces is equicontinuous at $x$ when, for every $\epsilon >0$, one $\delta >0$ works for all $f\in \mathcal{F}$: $d(x,y)<\delta$ implies $d(f(x),f(y))<\epsilon$.