Let $f$ be a function a $C^\infty(\mathbb{R}^n)$ function such that for all $x_2,...,x_n\in\mathbb{R}$, $f(.,x_2,...,x_n)$ goes to zero at infinity. By Rolle's theorem there exists $y\in\mathbb{R}$ such that $\partial_1f(y,x_2,...,x_n)=0$. The thing is that $y$ depends on $x_2,...,x_n$, is there a technique to show that $\varphi:x_2,...,x_n\mapsto y$ is continuous ?
It would be first necessary to have a procedure to select $y$ uniquely so that $\varphi$ is well-defined and then show that it is continuous (maybe by implicit function theorem).
Thanks!