Negative results for approximation using single layer and multilayer feedforward neural networks

We prove a negative result for the approximation of functions defined on compact subsets of $\mathbb{R}^d$ (where $d \geq 2$) using single layer feedforward neural networks with arbitrary continuous activation function. In a nutshell, this result claims the existence of target functions which 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 arbitrary number of layers, for activation functions which are either rational functions, or continuous splines with finitely many pieces.