A superposition theorem for unbounded continuous functions

Abstract:

Let ${R^n}$ be the n-dimensional Euclidean space. We prove that there are 4n real functions ${\varphi _q}$ continuous on ${R^n}$ with the following property: Every real function f, not necessarily bounded, continuous on ${R^n}$, can be written $f(x) = \Sigma _{q = 1}^{2n + 1}g({\varphi _q}(x)) + \Sigma _{q = 2n + 2}^{4n}h({\varphi _q}(x)),x \in {R^n}$, where g, h are 2 real continuous functions of one variable, depending on f.