Poly-time computability of inversion of poly-time real functions
In 1-dimensional case, the function has to be strictly-monotone. 
Given $y_n$ do a binary search to find $x_n$.

If $\mathsf{P} = \mathsf{NP}$, 
given $y_n$ we can guess $x_n$ and then verify that $f(x_n) = y_n$.
The dimension doesn't matter for this part.

For the other part, that is, if $INV_d$ is polytime then $\mathsf{P} = \mathsf{UP}$,
we can reduce it to $INV_2$ by noting that every 2-dimensional function 
can be extended to a $d$-dimensional one by letting other coordinates map as identity.
So for $d'&lt;d$,
$INV_d$ being polytime computable implies that $INV_{d'}$ is also polytime.
Therefore $INV_d$ being polytime implies that $INV_2$ is polytime and therefore $\mathsf{P}=\mathsf{UP}$.