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    $\begingroup$ So there is no requirement about when the double limits exist, or that they always both exist or both diverge, only that, when they both exist, they are equal? $\endgroup$ Commented Feb 20, 2025 at 13:31
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    $\begingroup$ @Lspice: exactly. IF both exist, THEN they are equal. $\endgroup$ Commented Feb 20, 2025 at 14:02
  • $\begingroup$ One tag of yours implicitly suggests the definition of a cardinal characteristic - the minimal cardinality $\kappa$ such that no $Y \in [2^\omega]^\kappa$ is DLC, let us call this number the double-limit discontinuity number $\mathfrak{d}\mspace{-1mu}\mathfrak{d}$. Bounding that from below by a run-of-the-mill cardinal characteristic would show that ZFC fails to refute Statement 1. $\endgroup$ Commented Aug 7, 2025 at 13:32