uniform continuity

Ordinary continuity lets the required closeness of inputs depend on where you are: near a steep part of the graph you must zoom in more. Uniform continuity is the stronger promise that one single setting of the input tolerance works everywhere at once — no matter where on the domain you stand, the same delta does the job.

Precisely, f is uniformly continuous on a set D if for every epsilon > 0 there is a delta > 0 such that, for all points x and y in D, |x - y| < delta implies |f(x) - f(y)| < epsilon. The crucial difference from plain continuity is the order of quantifiers: here delta is chosen before, and independently of, the points x and y, so it cannot adapt to a difficult location.

Uniform continuity implies continuity, but not the reverse. The classic gap: f(x) = x^2 is continuous on all of R yet not uniformly continuous there, because at large x even a tiny input gap produces a large output gap. The redeeming theorem (Heine–Cantor) is that a continuous function on a closed bounded interval — more generally on any compact set — is automatically uniformly continuous, so the distinction only bites on unbounded or non-closed domains.