I am interested in functions $$ f: \left| \begin{array}{rcl} \mathbb{R}^n & \longrightarrow &\mathbb{R}^n \\ \mathbf{x} & \longmapsto &(f_1(\mathbf{x}),\ldots,f_n(\mathbf{x})), \\ \end{array} \right.$$ such that the ordering of the coefficients of $\mathbf{x}$ is the same as the one of $f(\mathbf{x})$ for all $\mathbf{x} = (x_1,\ldots,x_n) \in \mathbb{R}^n$, i.e. such that, for any $i,j \in \{1,\ldots,n\}$, $$x_i \leq x_j \implies f_i(\mathbf{x}) \leq f_j(\mathbf{x}).$$
Is there a name for this?
This seems related to monotonicity. For instance, if $g: \mathbb{R} \rightarrow \mathbb{R}$ is an increasing function and $h: \mathbb{R}^n \rightarrow \mathbb{R}^+$ a non-negative function, the function $$f: \mathbf{x} = (x_1,\ldots,x_n) \mapsto (h(\mathbf{x})g(x_1),...,h(\mathbf{x})g(x_n)),$$ will satisfy the condition above. Maybe it is just implied by some multivariate notion of monotonicity ?