A family of functions fi:X → Y between metric spaces are equicontinuous at x in X if for any ε > 0 there exists δ > 0 such that whenever dX(x,y) < δ, then dY(fi(x), fi(y)) < ε for all i.
The family is uniformly equicontinuous if for any ε > 0 there exists δ > 0 such that whenever dX(x,y) < δ, then dY(fi(x), fi(y)) < ε for all i and all x,y.