Oscillation Revisited

TL;DR

This paper revisits the concept of oscillation of functions between metric spaces, extending previous work to establish a general joint continuity result for the oscillation function in the context of continuous functions.

Contribution

It formulates a general joint continuity theorem for the oscillation of functions, broadening the understanding of oscillation behavior in metric space mappings.

Findings

Established a joint continuity result for oscillation functions

Extended classical oscillation concepts to metric space mappings

Provided conditions under which oscillation is continuous for continuous functions

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 $\langle X,d \rangle$ and $\langle Y,\rho \rangle$ 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.