Let $\sigma : \mathbb{R} \to \mathbb{R}$ be a non-affine, continuous activation function that is continuously differentiable at some point with nonzero derivative there. Let $K \subset \mathbb{R}^m$ be compact. Then $\mathcal{N}^{\sigma}_{m,n,m+n+2}$ — networks with $m+n+2$ neurons per hidden layer and an arbitrary number of hidden layers — is dense in $C(K, \mathbb{R}^n)$ with respect to the uniform norm.