Let \( \sigma \) be any continuous sigmoidal function such that

$$ \sigma(z) = \left\{\begin{array}{cc} 1 & z\rightarrow \infty\\ 0 & z \rightarrow -\infty \end{array}\right. $$

Given a continuous and deterministic function \( F(\boldsymbol{x}) \) on the unit cube in \( d \)-dimensions \( F\in [0,1]^d \), \( x\in [0,1]^d \) and a parameter \( \epsilon >0 \), there is a one-layer (hidden) neural network \( f(\boldsymbol{x};\boldsymbol{\Theta}) \) with \( \boldsymbol{\Theta}=(\boldsymbol{W},\boldsymbol{b}) \) and \( \boldsymbol{W}\in \mathbb{R}^{m\times n} \) and \( \boldsymbol{b}\in \mathbb{R}^{n} \), for which

$$ \vert F(\boldsymbol{x})-f(\boldsymbol{x};\boldsymbol{\Theta})\vert < \epsilon \hspace{0.1cm} \forall \boldsymbol{x}\in[0,1]^d. $$