Composition of real functions

Question

Let $x,y\in\mathbb R^2$.

For any continuous function $f:(\mathbb R^2)^2\to\mathbb R$,

1. Does there always exist two continuous functions $g:\mathbb R^2\to\mathbb R$ and $h:\mathbb R^2\to\mathbb R$, such that $f(x,y)=h(g(x),g(y))$?
2. Does there always exist continuous functions $g_1:\mathbb R^2\to\mathbb R$, $g_2:\mathbb R^2\to\mathbb R$, and $h:\mathbb R^2\to\mathbb R$, such that $f(x,y)=h(g_1(x),g_2(y))$?

(2) seems trivally hold, but I don't know how to prove it. Any ideas? When considering a finite domain set instead of $\mathbb R$, (1) seems trivially hold, too.