Exercise 5, § 2

Let $U$ be an open neighbourhood of 0 in $\mathbf{C}^n$ and let $f_1, . . . , f_n$ be analytic mappings from U into $\mathbf{C}$. Let $f: U\rightarrow \mathbf{C}^n$ be the mapping given by $x\mapsto (f_1(x), . . . , f_n(x))$, let $J_f: U\rightarrow \mathbf{M}_n(\mathbf{C})$ be the mapping $x\mapsto (\partial_if_j(x))$ and let $h=$ det$\circ J_f$.

(a) Prove by induction on $n$ that $J_f$ is zero at every point of $\overset{-1}{h}(0)$. (b) Suppose that $f$ is injective; prove that $h$ does not vanish.