Let \(f:\mathbb {R}^n\to \mathbb {R}^n\) be a smooth function such that \(f(0)=0\). Show that the graph of \(f\)
\[ \operatorname {Gr}(f)=\{(x,f(x))\in \mathbb {R}^n\times \mathbb {R}^n\} \]
and the diagonal
\[ \Delta =\{(x,x)\in \mathbb {R}^n\times \mathbb {R}^n\} \]
intersect transversely at \((0,0)\) in \(\mathbb {R}^n\times \mathbb {R}^n\) if and only if the derivative \(f_*(0):T_0\mathbb {R}^n\to T_0\mathbb {R}^n\) does not have \(+1\) as an eigenvalue.