Can a scalar-valued multivariate function be invertible?

Question

Consider the scalar-valued multivariate function: $$ z = f(x,y) $$ Where $x,y,z \in \mathbb{R}$. If $f$ maps $\mathbb{R}^2$ to $\mathbb{R}$, can an inverse function $f^{-1}$ that maps $\mathbb{R}$ to $\mathbb{R}^2$ exist? If so, what are some examples?

Answer 1

If we do not impose any restriction on $f$, then the answer is Yes.

Recall that $\mathbb{R}$ and $\mathbb{R}^2$ have the same cardinalitty $c$, i.e., there exists a bijection $f:\mathbb{R}\rightarrow\mathbb{R}^2$.