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Oscillation Revisited

Abstract

In previous joint work by G. Beer and S. Levi, the authors studied the oscillation Ω(f, A) of a function f between metric spaces 〈X, d〉 and 〈Y, ρ〉 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 xX. The main purpose of this article is to formulate a general joint continuity result for (f, A)↦Ω(f, A) valid for continuous functions.

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Acknowledgments

The first author thanks the Auckland University of Technology for its hospitality in February 2016. The second author thanks the support of the National Natural Science Foundation of China, grant No. 11571158, and the paper was partially written when he visited Minnan Normal University in April 2016 as Min Jiang Scholar Guest Professor.

Author information

Correspondence to Gerald Beer.

Additional information

Dédié à Michel Théra pour son soixante-dixième anniversaire

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Cite this article

Beer, G., Cao, J. Oscillation Revisited. Set-Valued Var. Anal 25, 603–616 (2017). https://doi.org/10.1007/s11228-017-0425-8

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Keywords

  • Oscillation
  • Strong uniform continuity
  • UC-subset
  • Hausdorff distance
  • Locally finite topology
  • Finite topology
  • Strong uniform convergence
  • Very strong uniform convergence
  • Bornology

Mathematics Subject Classification (2010)

  • Primary 54E40
  • 54B20
  • 26A15
  • Secondary 54E35
  • 54C35