Prove that scalar functions of vectors cannot be inverted

Question

The following seems obvious to me (because information is clearly lost), but I have no idea how to prove it:

Suppose we have some arbitrary complex vector $\mathbf{A}$ with $m$ components. Let $f(\mathbf{A}):\mathbb{C}^{m}\rightarrow\mathbb{C}$. Prove that there does not exist a function $f$ for which we can retrieve $\mathbf{A}$ from $f(\mathbf{A})$ unless $m=1$.

Answer 1

The statement is false. Take a bijection of $\mathbb{R}^m$ with $\mathbb{R}$, and do this with both the real and complex components separately, to get a bijection between $\mathbb{C}^m$ and $\mathbb{C}$.

This is, of course, not linear -- there is no such linear $f$, by a simple dimensionality argument.