equicontinuity
Uniform continuity asks one function not to change too fast: a single delta works for all points. Equicontinuity raises the bar to an entire family of functions: ONE delta must work for all points AND all functions in the family simultaneously. It is a 'no function is too steep, and they are all gentle to the same degree' condition.
Definition (uniform equicontinuity on a set E): a family F of functions is equicontinuous if for every epsilon > 0 there exists delta > 0 such that for every f in F, |x - y| < delta implies |f(x) - f(y)| < epsilon. Crucially the same delta serves every member f of F. There is also a pointwise version (delta allowed to depend on the base point x but still not on f); on a compact set the two notions coincide.
Equicontinuity is the precise 'no wild wiggling' control that the Arzelà–Ascoli theorem pairs with uniform boundedness to guarantee a uniformly convergent subsequence. A standard sufficient condition: if every f in F satisfies a common Lipschitz bound |f(x) - f(y)| <= L|x - y| with the same L, the family is equicontinuous (take delta = epsilon/L). Without this control a family can be uniformly bounded yet have no convergent subsequence.
The family f_n(x) = sin(nx) on [0, 1] is uniformly bounded by 1 but NOT equicontinuous: f_n'(x) = n cos(nx) has slopes growing without bound, so no single delta tames them all. By contrast {x, x/2, x/3, ...} is equicontinuous (common Lipschitz constant 1). Only the equicontinuous, bounded family is forced to have a uniformly convergent subsequence.
sin(nx) is bounded but not equicontinuous; its slopes blow up.