[Math] Definition for monotonicity of multivariate function

Question

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!

Answer 1

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.