Suppose I have a function $f$ with domain $X$. Is there a theorem or some other known result which states that it is possible to: "divide $X$ into subdomains $X_i$, where $\cup_i X_i = X$, and define $g_i(x) = g(x)$ when $x \in X_i$ and zero elsewhere such that for all $i$, $g_i^{-1}(x)$ is unique or $g_i(x)$ is constant?" In this case I am envisioning that $\sum_i g_i(x) = g(x)$.
If this is not true in general, under what conditions of $g(x)$ can I guarantee that this is true?
Is this valid for multivariate functions? I.e. can I divide the domain of the multivariate function to subdomains such that in each subdomain, the inverse of the multivariate function is unique?