Consider a set of $n$ linearly independent functions on $\mathbb{R}^n$ i.e. $\{f_i:\mathbb{R}^n \rightarrow \mathbb{R}\}_{i=1}^n$. These functions are homogeneous polynomials.
Using $\{f_i\}$ define a coordinate transformation $T:\mathbb{R}^n \rightarrow \mathbb{R}^n$ defined by $T(x) = (f_1(x), \cdots, f_n(x))$.
How to prove that $T(\cdot)$ is a local homeomorphism except possibly at some isolated points?.
Thanks in advance.