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"monotonic" is well defined for functions $f(x)$, where e.g. $x\in[0,1]$ and $f(x)\in\mathbb{R}$. The quality I particularly care about is that if $f(x)$ is monotonic then it will not have any local extrema for $x\in(0,1)$.

Is there an analogous word for a function $g(x,y)$ with $x,y\in[0,1]$ and $g(x,y)\in\mathbb{R}$, where $g(x,y)$ has no local extrema for $x,y\in(0,1)$?

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"Inverse image of a contractible set is contractible" is what I have often thought the correct generalization of monotonicity to higher dimensions to be, and it fits your requested criteria. But I don't know a word for it. –  Will Sawin Apr 19 '12 at 21:53
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WHat about "open"? –  Igor Rivin Apr 19 '12 at 23:03
    
Constant functions are monotone but are only open in trivial cases. $\:$ Will's suggested criterion would never hold for (non-empty) disconnected domains and contractible ranges. $\:$ (I don't have my own suggestion at this point.) $\;\;$ –  Ricky Demer Apr 19 '12 at 23:31
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@Ricky: There is a good reason for that, I think. Any definition of monotonicity that a) depends only on topology and b) never says a function on a subset of the real line is monotonic when it isn't has that property. @Misha: I'll look it up. –  Will Sawin Apr 20 '12 at 2:58
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What about "harmonic for some conformal structure" ? –  BS. Apr 21 '12 at 9:52
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