A common form of the Stone–Weierstrass theorem is the statement that polynomial functions are dense under the $\ell_\infty$-norm in the space of continuous functions on $[0,1]^n$ for any $n<\omega$.

A trivial restatement of this fact is this: If we let $M= ([0,1],\dots)$ be the induced structure on the definable set $[0,1]$ as a subset of $\mathbb{R}$ as an ordered field, then for any continuous function $f:[0,1]^n\to [0,1]$ and any $\varepsilon > 0$, there is a definable function $g:M^n \to M$ such that $\left\lVert f-g\right\rVert < \varepsilon$. (Incidentally we don't actually need multiplication for this. The ordered group structure is enough.)

Since RCF is NIP, the induced structure on $[0,1]$ is NIP as well (and in fact o-minimal). I'm curious if this approximating property can be accomplished in a stable theory. I can think of more variations of this question than I should I put in an MO question, but I think the following two are reasonable to consider first.

> **Question 1:** Does there exist a structure $M$ whose underlying set is $[0,1]$, whose theory is stable, and which has the property that for any continuous function $f:[0,1]^n \to [0,1]$ and any $\varepsilon > 0$, there is a definable function $g : M^n \to M$ such that $\left\lVert f -g \right\rVert_\infty < \varepsilon$?

> **Question 2:** Assuming the first question has a positive answer, is there such a structure in which the witnessing $g$'s are continuous?

Note that the question doesn't depend on whether we interpret 'definable' as $\varnothing$-definable or definable with parameters.