On the representation of functions as a sum of several compositions

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 $n>2$.