A.A. Agrachev, On superpositions of continuous functions

Title: On superpositions of continuous functions
Authors: A.A. Agrachev
Journal title: Mathematical Notes
Year: 1974
Issue: 4
Volume: 16
Pages: 897–900
Citation:

A. A. Agrachev, “On superpositions of continuous functions”, Mat. Zametki, 16:4 (1974), 517–522

Abstract:

We show that if $\Phi$ is an arbitrary countable set of continuous functions of $n$ variables, then there exists a continuous, and even infinitely smooth, function $\psi(x_1, ..., x_n)$ such that $\psi(x_1, ..., x_n) \neq g [\phi (f_1(x_1), ..., f_n(x_n)]$ for any function $\phi$ from $\Phi$ and arbitrary continuous functions $g$ and $f_i$ depending on a single variable.