Representation of continuous functions of three variables by the superposition of continuous functions of two variables

Abstract

The present work is devoted to the proof of the following theorem, which was stated in an earlier note [1].

Theorem 1. Every real continuous function f(x ₁ ,x ₂ ,x ₃) of three variables, defined on the unit cube E ³ , can be represented in the form

$$ f(x_{1},x_{2},x_{3}) = \sum^{3}_{i=1} \sum^{3}_{j=1} h_{ij}[\varphi_{ij}(x_{1},x_{2}),x_{3}], $$

where h _ij and φ _ij are real continuous functions of two variables.