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In a function approximation with a neural network, an input dataset is mappedto an output index by optimizing the parameters of each hidden-layer unit. Fora unary function, we present constraints on the parameters and its secondderivative by constructing a continuum version of a one-hidden-layer neuralnetwork with the rectified linear unit (ReLU) activation function. The networkis accurately implemented because the constraints decrease the degrees offreedom of the parameters. We also explain the existence of a smooth binaryfunction that cannot be precisely represented by any such neural network.

No one-hidden-layer neural network can represent multivariable functions