A superposition theorem for unbounded continuous functions

Abstract

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