Denseness of a preimage. Continous and surjective function Let $F:\mathbb{R^2} \to \mathbb{R}$ surjective and continuous on $\mathbb{R^2} $.
Let  $A\subset \mathbb{R}$, where $A$ is a dense set on $\mathbb{R}$.
Is $F^{-1}(A)$ a dense set on $\mathbb{R^2}$ ?

Let $F(x,y)=xy$ if $x>0$ and $F(x,y)=0$ if $x \leq 0$. Let $A$ be any dense set in $\mathbb R$ which does not contain $0$, say non-zero rationals. Then $F^{-1}(A)$ does not intersect $\{(x,y): x \leq 0\}$ so it is not dense. 
