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

We prove, for $d\geq 2$, a negative result for approximation of functions defined con compact subsets of $\mathbb{R}^d$ with single layer feedforward neural networks with arbitrary activation functions. In philosophical terms, this result claims the existence of learning functions $f(x)$ which are as difficult to approximate with these neural networks as one may want. We also demonstrate an analogous result (for arbitrary $d$) for neural networks with an arbitrary number of layers, for some special types of activation functions.