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Sep 7, 2022 at 15:19 comment added Philippe Gaucher @AndreasBlass You're perfectly right. I use strictly increasing only to prove specific properties of the "topologification" of $f$.
Sep 7, 2022 at 14:56 comment added Andreas Blass This theorem doesn't use that $f$ is strictly increasing. It treats each component $f_i$ independently of the others, and it amounts to saying simply that $f_i$ is a weakly increasing function to $\{0,1\}$. That is, the max and min in the formula for $f_i$ just say that $f_i(\vec x)=1$ iff $\vec x\geq\vec\varepsilon$ for some $\vec\varepsilon$ with $f_i(\vec\varepsilon)=1$.
Sep 7, 2022 at 7:46 vote accept Philippe Gaucher
Sep 7, 2022 at 7:38 history answered Philippe Gaucher CC BY-SA 4.0