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