I'd like to know whether there is an example of universal approximators whose value at the origin is fixed as zero.
Motivation
Let's think about $C_{0} := \{f:X \to \mathbb{R}| f \text{is continuous}, f(0)=0\}$, where $X \subset \mathbb{R}^n$ is a compact set. One may use a universal approximator $\hat{f}$ (e.g., feedforward neural network) to approximate any function $f$ in $C_{0}$ for every $\epsilon > 0$ so that $$ \lVert \hat{f} - f \rVert_{\infty} < \epsilon. $$ But one cannot guarantee that $\hat{f}(0) = 0$.
Question
Given $f \in C_0$, for every $\epsilon > 0$, is there any universal approximator $\hat{f}_{0}$ such that $\lVert \hat{f}_0 - f\rVert_{\infty} <\epsilon$ and $\hat{f}_0(0) = 0$?
Notes
EDIT: I'd like to know the name of $C_{0}$ or such universal approximator $\hat{f}_0$ if it exists.