TITLE: Is this true for a continuously differentiable function
QUESTION [6 upvotes]: Prove or find a counterexample: if $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is continuously differentiable with $f(0)=0$  then there exist continuous functions $g_1,\dots,g_n:\mathbb{R}^n\rightarrow \mathbb{R}$ with
$$f(x)=x_1g_1(x_1,\dots, x_n)+\dots+x_ng_n(x_1,\dots, x_n)$$
I don't know where to start. Should I use the implicit function theorem?
Please give me some hints.
Happy new year.

REPLY [4 votes]: Fix $x\in{\mathbb R}^n$ and consider the auxiliary function $$\phi(t):=f(t\,x)\qquad(0\leq t\leq 1)\ .$$
The chain rule then gives
$$f(x)=\phi(1)-\phi(0)=\int_0^1\phi'(t)\>dt=\int_0^1\nabla f(t\,x)\cdot x\>dt=\sum_{i=1}^n x_i\,g_i(x)$$
with
$$g_i(x):=\int_0^1 f_{.i}(t\, x)\>dt\qquad(1\leq i\leq n)\ .$$