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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 $\langle X,d \rangle$ and $\langle Y,ρ\rangle$ 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 $x \in X$. The main purpose of this article is to formulate a general joint continuity result for $(f,A) \mapsto Ω(f,A)$ valid for continuous functions.