Let $C({ \mathbb R}^n)$ denote the set of real valued continuous functions defined on ${ \mathbb R}^n$. We prove that for every $n\ge 2$ there are positive numbers $\lambda _1 , \ldots , \lambda _n$ and continuous functions $\phi_1 ,\ldots , \phi _m \in C({ \mathbb R})$ with the following property: for every bounded and continuous $f\in C( { \mathbb R}^n )$ there is a continuous function $g\in C({ \mathbb R} )$ such that $$f(x)=\sum_{ q=1}^m g\left( \sum_{ p=1}^n \lambda _p \phi _q (x_p ) \right)$$ for every $x=(x_1 ,\ldots , x_n )\in { \mathbb R}^n$. Consequently, every $f\in C({ \mathbb R}^n)$ can be obtained from continuous functions of one variable using compositions and additions.
Categories: math.CA