Oscillation Revisited

In previous work by Beer and Levi [8, 9], the authors studied the oscillation Ω(f, A) of a function f between metric spaces hX, di and hY, ρi at a nonempty subset A of X, defined so that when A = {x}, we get Ω(f, {x}) = ω(f, x), where ω(f, x) denotes the classical notion of oscillation of f at the point x ∈ X. The main purpose of this article is to formulate a general joint continuity result for (f, A) 7→ Ω(f, A) valid for continuous functions