The question I want to pose is the following
assume $f:\mathbb R^p\times \mathbb R\times \mathbb R\to \mathbb R$, $(w,x,y)\mapsto f(w,x,y)$ is such that in a neighborhood $U$ of the origin it holds that both
$xf\in C^\infty(U)$,
$yf\in C^\infty(U)$.
Prove then that $f\in C^\infty (U)$.
The fact is that I encountered this problem as a technical lemma in a paper to a subsequent theorem, and it is stated that the proof is trivial and therefore left to the reader. I was unsuccesful in proving this so I am asking for an help.
Bonus Question... The previous statement remains true if we substitute smooth with analytic.
Thanks for the help.