Could you prove or disprove the following statement?
Let $f\colon[0,1]^2\rightarrow \mathbb R$ be a continuous function. Then there are continuous functions $g,\ h\colon [0,1]\rightarrow \mathbb R$ and $\Phi\colon \mathbb R \to \mathbb R$ such that $$ f(x,y) = \Phi(g(x) + h(y)).$$
(This problem popped up in my mind while I was thinking about this related one on MO. I couldn't find an easy proof or a disproof. This version is much weaker than the one asked at MO, since $g$ and $h$ do depend on $f$ here.)