equicontinuity

A single continuous function promises that small moves in the input cause small moves in the output. Equicontinuity is the same promise made simultaneously, with one common rule, by a whole family of functions at once. It is the condition that stops a sequence of functions from secretly getting steeper and steeper as you go along — the wiggling is reined in uniformly across the family, which is exactly what is needed to extract a convergent subsequence.

Precisely, a family F of functions on a set is equicontinuous at a point x_0 if for every error tolerance epsilon > 0 there is a single distance delta > 0 — the SAME delta for every f in F — such that |x - x_0| < delta guarantees |f(x) - f(x_0)| < epsilon for every f in the family. If one delta works at every point (often via a common Lipschitz bound) the family is uniformly equicontinuous. Contrast this with ordinary continuity, where each function gets its own delta, and that delta is allowed to shrink to nothing as you scan across the family. The force of equicontinuity is the word SAME: one delta must serve everybody. A clean sufficient condition: if all the f have a common bound on their derivatives, |f'| <= L, then |f(x) - f(y)| <= L|x - y| for all of them, so the single delta = epsilon/L works — derivative-bounded families are equicontinuous.

Equicontinuity is the subtle half of the Arzela-Ascoli theorem and therefore the hidden ingredient in every normal-family argument. In complex analysis you almost never check it by hand: the Cauchy estimates convert a bound on the size of holomorphic functions directly into a bound on their derivatives, which hands you equicontinuity automatically — this is precisely the step that makes Montel's theorem work. Honest caveat: a family can be pointwise bounded and each member continuous, yet fail to be equicontinuous (think of bumps that grow steeper), and then no convergent subsequence need exist. Equicontinuity, not mere individual continuity, is what compactness requires.