On the Representation of Continuous Functions of Several Variables as Superpositions of Continuous Functions of one Variable and Addition

The aim of this paper is to present a brief proof of the following theorem: Theorem. For any integer n ≥ 2 there are continuous real functions ψ p q (x) on the closed unit interval E 1 = [0;1] such that each continuous real function f(x 1 ,…,x n ) on the n-dimensional unit cube E n is representable as $$f\left( {{{x}_{1}}, \ldots ,{{x}_{n}}} \right) = \sum\limits_{{q = 1}}^{{q = 2n + 1}} {Xq\left[ {\sum\limits_{{p = 1}}^{n} {{{\psi }^{{pq}}}\left( {{{x}_{p}}} \right)} } \right]} ,$$ (1) where x q (y) are continuous real functions.