Does $\nabla u$ have null tangential component to $\mathrm{supp} \ u$?

Question

Let $u: \mathbb{R}^N \to \mathbb{R}$ be a smooth function. Does $\nabla u$ have null tangential component to $\mathrm{supp}\, u $ ? That is, do we have $$\frac{\partial u}{\partial \nu} = 0 \Rightarrow |\nabla u| = 0$$ on $\mathrm{supp} \ u$, where $\nu $ denotes the outer normal to $\mathrm{supp} \ u$?

Answer 1

$u$ is smooth on all of $\Bbb R^n$, including on the boundary of $\text{supp }u$. Outside of $\text{supp }u$, we obviously have $\nabla u = 0$. Therefore by continuity, we have $\nabla u = 0$ on the boundary of $\text{supp }u$ as well.