The following result can be found in the paper "Natural operations on differential forms" by Richard S. Palais:

Corollary 4.3: "Let $f_{1}, \ldots , f_{n}$ be $n$ real valued functions of $n$
real variables. Necessary and sufficient conditions that
the mapping $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$
defined by $f(x) = (f_{1}(x), \ldots , f_{n}(x))$
be a diffeomorphism of $\mathbb{R}^{n}$ onto itself are:

(1) $\det (df_{i}/dx_{j})$ never vanishes.

(2) $\lim_{||x||\rightarrow \infty} ||f(x)|| = \infty$."

If we assume that each of the $f_{i}$:s are smooth functions also of some other variables in $y \in \mathbb{R}^{m}$, is satisfaction of the above conditions for all $y \in \mathbb{R}^{m}$ necessary and sufficient for the $x_{i}$:s to be functions of $y$ in the entire $\mathbb{R}^{n}$ (intuitively the answer is yes I suppose)?

Also, if $f$ is smooth in $x$ and $y$, will $x$ be smooth as a function of $y$?

(I just realized my question is related to a previous one: How to "globalize" the inverse function theorem?)

Best regards

Olav