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**MattMan** Let $x\in\partial X$ be a boundary point. Show that there exists a smooth nonnegative function $f$ on some open neighborhood $U$ of $x$, such that $f(z)=0$ if and only if $z\in\partial U$, and if $z\in\partial U$, then $df_z(\vec{n}(z))>0$.

Note that in this context $\vec{n}$ is the outward unit normal.

I was assigned this for homework, but I'm pretty sure it's not possible. In a basic example, this is saying that there is an $f:\R\to\R$ such that $f(x)>0$ for all $x\in (0,\infty)$ and $f(0)=0$, and $f'(0)<0$ (since the unit normal is -1). But this isn't possible, for that says that as $x\to 0^+$, that f(x) is increasing, which clearly can't be the case for nonnegative functions.

Is my logic wrong somewhere? Any help would be appreciated, thanks!