I've seen a couple of texts, including this answer on another question mention that any continuous function on the Long Line must eventually be constant, but I've not seen the reasoning as to why.

Why must this be true?

  • $\begingroup$ Have you seen the proof that a continuous (real-valued) function on the uncountable ordinal $\omega_1$ is eventually constant? It's basically the same idea. $\endgroup$ – Nate Eldredge Mar 21 at 15:45
  • $\begingroup$ @NateEldredge No. $\endgroup$ – Shufflepants Mar 21 at 15:49
  • $\begingroup$ @NateEldredge A proof can be found here e.g. if you assume the pressing down lemma. $\endgroup$ – Henno Brandsma Mar 21 at 17:13
  • $\begingroup$ @HennoBrandsma I'm afraid I don't really understand that proof. I can't see how the pressing down lemma prevents a function such as sin(x) being defined over the entire domain, remaining continuous, and never ending up constant. And if it can't, at what point must sin(x) become constant? $\endgroup$ – Shufflepants Mar 21 at 20:24
  • $\begingroup$ @Shufflepants you cannot define $\sin(x)$ on the long line. What is $\sin(\alpha, t)$ for an ordinal $\alpha$ and $t\in [0,1)$ in a continuous way? $\endgroup$ – Henno Brandsma Mar 21 at 20:27

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