Find a function $f : \mathbb{R}^3 \to \mathbb{R}^3$ with $f(0,0,0) = (1,2,3)$ that is locally invertible around $(0,0,0)$ but not globally invertible on $\mathbb{R}^3$.
I am struggling to find one that is not injective. I am also wondering if there is a way to show that $f$ is locally invertible without computing the determinant of $3×3$ matrix.
Such function $f : \Bbb R \to \Bbb R$ with $f(0) = 0$ is easy to construct. Now take $$(x,y,z) \mapsto (f(x)+1,y+2,z+3)$$