General approximation theorem on feedforward networks

We show that standard feedforward neural networks with as few as a single hidden layer and arbitrary bounded nonlinear (continuous or noncontinuous) activation functions which have two unequal limits in infinities can uniformly approximate (in contrast to approximate measurably) arbitrary bounded continuous mappings on Rⁿ with any precision. Especially, in a compact set of Rⁿ, standard feedforward neural networks with as few as a single hidden layer and arbitrary bounded nonlinear (continuous or noncontinuous) activation functions can uniformly approximate arbitrary continuous mappings with any precision. These results also hold for multi-hidden layer standard feedforward neural networks. We found that the boundedness and unequal limits at infinities conditions on the activation functions are sufficient, but not necessary