The question is: Give an example of a function $f$, continuous on $S=(0,1) \times (0,1)$, such that $f(S)=\mathbb{R}^2$.
I'm getting stuck on $f(x)=\tan\left(\pi \left(x-\frac{1}{2}\right)\right)$, which covers all of $\mathbb{R}$ but not all of $\mathbb{R}^2$.