Is any $C_c^\infty(\mathbb R)$-function the product of two $C_c^\infty(\mathbb R)$-functions?

Question

Let $f\in C_c^\infty(\mathbb R)$. Are we able to write $$f=gh$$ for some $g,h\in C_c^\infty(\mathbb R)$? Unfortunately, I've no idea how to I could prove or disprove this.

Answer 1

There exists a smooth compactly supported function $g$ that is $1$ on an open ball containing the support of $f$. Then $f=fg$.

For the definition of $g$: let $h(r)=0$ if $r \leq 0$ and $h(r)=e^{-1/r}$ if $r > 0$. It is well-known that $h$ is smooth. Let $h_1(r)=\frac{h(r)}{h(r)+h(1-r)}$: $h_1$ is smooth, zero on $(-\infty,0]$ and $1$ on $[1,\infty)$.

Take now $g(x)=h_1\left(2-\frac{|x|^2}{R^2+1}\right)$ where $\{f \neq 0\} \subset \mathcal{B}(0,R)$.