Uniform Continuity

Abstract

lim j→∞ xij = z. Let f : X → Y be a function with Y a metric space with metric ρ. We say that f is continuous at a point x0 ∈ X provided that for every > 0 there is a δ > 0 such that for every x ∈ X, if d(x, x0) < δ, then ρ(f(x), f(x0)) < . A function f : X → Y is said to be continuous provided that it is continuous at each point x ∈ X. A function f : X → Y is said to be uniformly continuous provided that for every > 0, there is a δ > 0 such that for every pair of points x and x′ in X, with d(x, x′) < δ, ρ(f(x), f(x′)) < . For a function f : X → Y to be uniformly continuous is stronger than being continuous.