Geodesic
On the representation of functions as a sum of several compositions
Sbornik. Mathematics, Tome 74 (1993) no. 1, pp. 119-130 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\varphi_i$ be continuous mappings of a compactum $X$ onto compacta $Y_i$, $i=1,\dots,n$. The following theorem is known for $n=2$: if any bounded function $f$ on $X$ can be represented in the form $f=g_1\circ\varphi_1+g_2\circ\varphi_2$, where $g_1$ and $g_2$ are bounded functions on $Y_1$ and $Y_2$, then any continuous $f$ can be represented in the same form with continuous $g_1$ and $g_2$. An example is constructed showing that the analogous theorem is false for $n>2$. 
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V. A. Medvedev. On the representation of functions as a sum of several compositions. Sbornik. Mathematics, Tome 74 (1993) no. 1, pp. 119-130. http://geodesic.mathdoc.fr/item/SM_1993_74_1_a9/

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