Consider the domain of all functions that will be mentioned is $X$, a compact subspace of $\mathbb{R}^n$.
Suppose that, given a continuous function $f$, $\forall \epsilon > 0$, $\exists \hat{f} \in \mathcal{F}$ such that $\lVert f - \hat{f} \rVert_{\infty} < \epsilon$.
Then, for any nonnegative continuous function $g$, for every $\epsilon > 0$, does there exist $\hat{f} \in \mathcal{F}$ such that $\lVert g - \max(0, \hat{f})\rVert_{\infty} < \epsilon$?
That is, when we have a universal approximator $\mathcal{F}$ for continuous functions, can we approximate any nonnegative function by the form of $\max(0, \hat{f})$ where $\hat{f} \in \mathcal{F}$?
Please see the comments; I don't know whether it is possible to change this question as answered without an answer, so I posted this answer.