Our metric space is $X = [0,1]^n$. Consider a set of points $S \subset X$ such that for any $x, x' \in S, x'_i > x_i \implies x'_j \leq x_j$ for some $j \leq n$. $S$ can be uncountable. My question is, for any such $S$, does there exist a function $\phi: [0,1]^n \rightarrow \mathbb{R}$ with $\phi$ is increasing (weakly) in each of its components, such that $S \subseteq L$ for some level set of $\phi$?
My question is related to this and this question. Though since in my case $S$ can be uncountable, one of them comes very close to answering it but doesn't.
Edit: A previous version of this question said "analytic function". I don't need the function to be analytic. It was a mistake. I misunderstood analytic functions. My apologies.