The general approximation problem of interest to the area of feedforward neural net works is stated. Solutions for som e special cases are given. which include an upper bound on the number ofnodes in hidden layerts ) and the weights for that confi guration. Analytical solutions to the generalfeedfo rward neural network problem in one-dimensional cases requiring an infinite number of nodes are presented. The practical solutio ns (not requiring an infinite number ofnodes) in one-dimensional cases are derived under piecewise constant approximat ions with constant width partitions, under piecewiseconstantapproximations with variablewidth partitions. and under piecewise linear approximations using ramps instead of sigmoids. A four layer solution to the general feedfo rward neural network problem in the n-dimensional case is presented. A three layer solution to the general feedforward neural network problem in the n-dimensional case with piecewise constant approximation requires the use of the corner fun ction as the activation fu nction. The cornerf unction. a special case ofn dimensional sigmoid function, isfound to have desirable characteristics. and can be used to approximate fu nctions with much weaker requirements (only boundedness and piecewise continuity.) Concaveregions can be formed with a single layer ofnodes with the corner function.