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I've seen in multiple posts (e.g. here, here and here) on how continuity properties of functions imply Borel-Measurability, in a very natural way relating the topological definition of continuity with the definition of Borel-Measurability itself. I'm wondering if everyone knows about results in the opposite direction: what can we say about continuity properties (e.g. continuous, almost surely continuous, countably-discontinuous, or even other definitions of continuity such as equicontinuity, and so on) by knowing that a function is Borel Measurable?

Heatconomics
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    Not so much I am afraid. Very weird functions can still be Borel measurable. – drhab Nov 21 '19 at 12:59
  • Thanks! Do you know if any additional conditions would help? Such being bounded, having compact support, and so on?@drhab – Heatconomics Nov 21 '19 at 13:20
  • Not really. I never dived into this material because it looked so hopeless :-). – drhab Nov 21 '19 at 13:25
  • note that the Dirichlet function is nowhere continuous and Borel measurable. I dont think we can say something useful because if we reject the axiom of choice then any function will be Borel measurable –  Nov 21 '19 at 14:34
  • @Masacroso that is true! As drhab said, it seems pretty much hopeless without constraining assumptions. Thanks y'all – Heatconomics Nov 21 '19 at 14:36

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