This paper by G. Cybenko 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 results settle an open question about the representability in the class of single hidden layer neural networks. The paper shows that arbitrary decision regions can be arbitrarily well approximated by continuous feedforward neural networks with only a single internal, hidden layer and any continuous sigmoidal nonlinearity. It also discusses approximation properties of other possible types of nonlinearities that might be implemented by artificial neural networks. The main result of the paper is that sums of the form $ G(x) = \sum_{j=1}^{N} \alpha_j \sigma(y_j^T x + \theta_j) $ are dense in the space of continuous functions on the unit cube if $ \sigma $ is any continuous sigmoidal function. This result is proven using functional analysis techniques, including the Hahn–Banach and Riesz Representation Theorems. The paper also shows that any continuous sigmoidal function is discriminatory, meaning that if the integral of $ \sigma(y^T x + \theta) $ over the unit cube is zero for all $ y $ and $ \theta $, then the measure must be zero. The paper applies these results to artificial neural networks, showing that networks with one internal layer and an arbitrary continuous sigmoidal function can approximate continuous functions with arbitrary precision. It also discusses the approximation of decision functions for general decision regions, showing that any finite measurable partition of the unit cube can be approximated by such networks. The paper also considers other activation functions with similar approximation properties, including discontinuous sigmoidal functions, sine and cosine functions, and exponential functions. It concludes that continuous sigmoidal functions are discriminatory and that any continuous function can be uniformly approximated by a continuous neural network with one internal, hidden layer and an arbitrary continuous sigmoidal nonlinearity.This paper by G. Cybenko 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 results settle an open question about the representability in the class of single hidden layer neural networks. The paper shows that arbitrary decision regions can be arbitrarily well approximated by continuous feedforward neural networks with only a single internal, hidden layer and any continuous sigmoidal nonlinearity. It also discusses approximation properties of other possible types of nonlinearities that might be implemented by artificial neural networks. The main result of the paper is that sums of the form $ G(x) = \sum_{j=1}^{N} \alpha_j \sigma(y_j^T x + \theta_j) $ are dense in the space of continuous functions on the unit cube if $ \sigma $ is any continuous sigmoidal function. This result is proven using functional analysis techniques, including the Hahn–Banach and Riesz Representation Theorems. The paper also shows that any continuous sigmoidal function is discriminatory, meaning that if the integral of $ \sigma(y^T x + \theta) $ over the unit cube is zero for all $ y $ and $ \theta $, then the measure must be zero. The paper applies these results to artificial neural networks, showing that networks with one internal layer and an arbitrary continuous sigmoidal function can approximate continuous functions with arbitrary precision. It also discusses the approximation of decision functions for general decision regions, showing that any finite measurable partition of the unit cube can be approximated by such networks. The paper also considers other activation functions with similar approximation properties, including discontinuous sigmoidal functions, sine and cosine functions, and exponential functions. It concludes that continuous sigmoidal functions are discriminatory and that any continuous function can be uniformly approximated by a continuous neural network with one internal, hidden layer and an arbitrary continuous sigmoidal nonlinearity.