Let $f(x):\mathbb R^n \to \mathbb R$ be a smooth function, and let $f(0)=0$.
I alway see that someone rewrite the function in the form $f(x)=x_1b_1(x_1)+x_2b_2(\overline {x_2})+...+x_nb_n(\overline x)$,
where the $\overline {x_i}=[x_1\ \ x_2\ \ ...\ \ x_i]^\mathrm{T}$
Can someone provide a proof?