# Definition for monotonicity of multivariate function

Is there any standard definition for monotonicity of a multivariate function?

I suppose it's something like:

$\forall i: x_i \leq x_i' \implies f(x_1, \ldots, x_i, \ldots, x_k) \leq f(x_1, \ldots, x_i', \ldots, x_k)$

thanks!

• Can you actually clarify where the $\forall$ is being applied in your definition? I don't think it captures what should be intuitively monotone. Sep 4 '13 at 15:54
• Should there be primes on $x_1$ and $x_k$ on the right hand side? If so, our definitions agree. Sep 4 '13 at 16:24
• no, no primes, just 1 larger argument.. Sep 6 '13 at 7:11
• I think in terms of monotonicity, my definition is more standard. I don't know what else you would like to know here. Sep 8 '13 at 19:50

A sensible extension of monotonicity is the following. Let $A$ and $B$ be partially ordered sets. Let $f\colon A\rightarrow B$. $f$ is monotone if for each $x,y\in A$ s.t. $x\leq y$ we have that $f\left(x\right)\leq f\left(y\right)$. Just take $A=\mathbb{R}^{n}$ and $B=\mathbb{R}^{m}$ for the case you are interested in.
• But this would require the $\leq$ relation to be defined on tuples from A and on those from B? Sep 4 '13 at 16:19
• We can easily define the partial ordering by: $\mathbf{x} \leq \mathbf{y}$ if $\forall n : x_n \leq y_n$ Jun 17 '14 at 10:07
• With that the definition of @par is equivalent to your definition, Xaero182, since it follows from applying your definition multiple times on different $x_n$. Jun 17 '14 at 10:11