Oscillation Revisited

Abstract

In previous work by Beer and Levi [8, 9], the authors studied the oscillation $\Omega (f,A)$ of a function $f$ between metric spaces $⟨X,d⟩$ and $⟨Y,\rho ⟩$ at a nonempty subset $A$ of $X$, defined so that when $A=\{x\}$, we get $\Omega (f,\{x\})=\omega (f,x)$, where $\omega (f,x)$ denotes the classical notion of oscillation of $f$ at the point $x\in X$. The main purpose of this article is to formulate a general joint continuity result for $(f,A)\mapsto \Omega (f,A)$ valid for continuous functions.

13 pages