A note on the representation of continuous functions by linear superpositions

We consider the problem of the representation of real continuous functions by linear superpositions $\sum_{i=1}^{k}g_{i}\circ p_{i}$ with continuous $g_{i}$ and $p_{i}$. This problem was considered by many authors. But complete, and at the same time explicit and practical solutions to the problem was given only for the case $k=2$. For $k>2$, a rather practical sufficient condition for the representation can be found in Sternfeld [17] and Sproston, Strauss [16]. In this short note, we give a necessary condition of such kind for the representability of continuous functions.