  The universal approximation property (UAP) of neural networks is fundamental
for deep learning, and it is well known that wide neural networks are universal
approximators of continuous functions within both the $L^p$ norm and the
continuous/uniform norm. However, the exact minimum width, $w_{\min}$, for the
UAP has not been studied thoroughly. Recently, using a
decoder-memorizer-encoder scheme, \citet{Park2021Minimum} found that $w_{\min}
= \max(d_x+1,d_y)$ for both the $L^p$-UAP of ReLU networks and the $C$-UAP of
ReLU+STEP networks, where $d_x,d_y$ are the input and output dimensions,
respectively. In this paper, we consider neural networks with an arbitrary set
of activation functions. We prove that both $C$-UAP and $L^p$-UAP for functions
on compact domains share a universal lower bound of the minimal width; that is,
$w^*_{\min} = \max(d_x,d_y)$. In particular, the critical width, $w^*_{\min}$,
for $L^p$-UAP can be achieved by leaky-ReLU networks, provided that the input
or output dimension is larger than one. Our construction is based on the
approximation power of neural ordinary differential equations and the ability
to approximate flow maps by neural networks. The nonmonotone or discontinuous
activation functions case and the one-dimensional case are also discussed.
