Consider functions $f_i(x)$ that map $\mathbb{R}^n$ to $\mathbb{R}$ for $i\in{1,\dots,k}$. Assume these functions are monotonically increasing in their arguments and continuous everywhere Also, one can assume that they are bounded. Consider the set of inequalities $f_i(x)\leq 0$. What are some properties of the feasible set of this set of inequalities defined as $$\mathcal{F}=\bigcap_{1\leq i\leq k}\{x~|~f_i(x)\leq 0,~\forall i\}$$
For instance, consider the case of only one function ($k=1$). Let $x=(x_1,\dots,x_n)$ be a feasible point such that $f_1(x)$. Then define the set $$\mathcal{S}_x=\{y~|~y_j\leq x_j\}$$ Essentially $\mathcal{S}_x$ is set of all vectors that are component-wise lesser than $x$. This is a convex set. Moreover, due to monotonicity of $f_1(.)$, all points in $S_x$ are feasible as well, i.e. $\mathcal{S}_x\subset\mathcal{F}$. In fact, we can define the set $$\mathcal{A}=\{x~|~f_1(x)=0\}$$ It is easy to see that the set $$\mathcal{B}=\bigcup_{x\in \mathcal{A}}\mathcal{S}_x$$ is also a feasible set ($\mathcal{B}\subset\mathcal{F}$). Due to monotonicity of $f_1(.)$, we can prove that $\mathcal{B}$ is the only feasible set ($\mathcal{F}\subset\mathcal{B}$) . However, I am not able to generalize this kind of arguments beyond one function ($k>1$). Appreciate any help in this direction
