Consider the map $f: \mathbf{R}^{3} \rightarrow \mathbf{R}^{3}$ given by

$$f(x, y, z)=(x+y+z, x y+y z+z x, x y z)$$

Show that $f$ is differentiable everywhere and find its derivative.

Stating carefully any theorem that you quote, show that $f$ is locally invertible near a point $(x, y, z)$ unless $(x-y)(y-z)(z-x)=0$.