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?
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1Yes, since $\mathbb{R}$ and $\mathbb{R}^2$ have the same cardinality. – Vercingetorix Apr 28 '21 at 15:36
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Do you have an example in mind? – mhdadk Apr 28 '21 at 15:37
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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 $f:\mathbb{R}\rightarrow\mathbb{R}^2$.
Danny Pak-Keung Chan
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