G. Cybenko's paper "Approximation by Superpositions of a Sigmoidal Function" demonstrates that finite linear combinations of compositions of a fixed univariate function and a set of affine functionals can uniformly approximate any continuous function of \( n \) real variables with support in the unit hypercube. The key result shows that arbitrary decision regions can be arbitrarily well approximated by continuous feedforward neural networks with a single hidden layer and any continuous sigmoidal nonlinearity. The paper discusses the approximation properties of other types of nonlinearities that might be implemented by artificial neural networks and provides a detailed proof of the main theorem using functional analysis techniques. The results settle an open question about the representability of functions in the class of single hidden layer neural networks and have significant implications for signal processing and control applications.G. Cybenko's paper "Approximation by Superpositions of a Sigmoidal Function" demonstrates that finite linear combinations of compositions of a fixed univariate function and a set of affine functionals can uniformly approximate any continuous function of \( n \) real variables with support in the unit hypercube. The key result shows that arbitrary decision regions can be arbitrarily well approximated by continuous feedforward neural networks with a single hidden layer and any continuous sigmoidal nonlinearity. The paper discusses the approximation properties of other types of nonlinearities that might be implemented by artificial neural networks and provides a detailed proof of the main theorem using functional analysis techniques. The results settle an open question about the representability of functions in the class of single hidden layer neural networks and have significant implications for signal processing and control applications.