¹⁴When an arbitrary number of hidden layers is allowed, one can devise many alternative schemes. For instance, a squaring function x → x² can be approximated on a small interval via linear combinations of an arbitrary nonlinear function , since a Taylor expansion around a constant c gives x² = 2[(c + x) -(c) - x′(c)]∕′′(c) + O(x³). The only provision here is that is at least three times differentiable (or at least four times differentiable if we would have used the more accurate alternative x² = [(c + x) - 2(c) + (c-x)]∕′′(c) + O(x⁴)). These requirements are satisfied by our C^∞ functions ₀, ₁ and ₂. A multiplication xy can subsequently be constructed as a linear combination of squaring functions through xy = [(x + y)² - (x-y)²], xy = [(x + y)² -x² -y²] or xy = -[(x-y)² -x² -y²]. A combination of additions and multiplications can then be used to construct any multidimensional polynomial, which in turn can be used to approximate any continuous multidimensional function up to arbitrary accuracy. See also [33].