  In a number of papers, Y. Sternfeld investigated the problems of
representation of continuous and bounded functions by linear superpositions. In
particular, he proved that if such representation holds for continuous
functions, then it holds for bounded functions. We consider the same problem
without involving any topology and establish a rather practical necessary and
sufficient condition for representability of an arbitrary function by linear
superpositions. In particular, we show that if some representation by linear
superpositions holds for continuous functions, then it holds for all functions.
This will lead us to the analogue of the well-known Kolmogorov superposition
theorem for multivariate functions on the $d$-dimensional unit cube.
