  We prove a negative result for the approximation of functions defined on
compact subsets of $\mathbb{R}^d$ (where $d \geq 2$) using feedforward neural
networks with one hidden layer and arbitrary continuous activation function. In
a nutshell, this result claims the existence of target functions that are as
difficult to approximate using these neural networks as one may want. We also
demonstrate an analogous result (for general $d \in \mathbb{N}$) for neural
networks with an \emph{arbitrary} number of hidden layers, for activation
functions that are either rational functions or continuous splines with
finitely many pieces.
