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I apologize in advance if this has been asked elsewhere, but I couldn't find it. This seems like it should be a pretty simple question, but I'm drawing a blank.

If you know that $f(x) \sim g(x)$, then under what conditions can you show that $F(f(x)) \sim F(g(x))$?

(This is not homework; just for personal interest.)

Lowell
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  • Being uniformly continuous is not enough: consider $f(n)=\pi n$ and $g(n)=\pi n+\pi/2$, hence $f\sim g$, but if we consider $F(n)=\sin(n)$, we conclude $F(f)\nsim F(g)$. – MrSelberg Mar 17 '15 at 08:44
  • Yeah, that's pretty much the line of thought that made me ask the question. Maybe monotonicity gets around those issues? – Lowell Mar 17 '15 at 08:49
  • Wait, that makes no sense. Ignore my last comment. – Lowell Mar 17 '15 at 09:07

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