Is there a name of such functions? Let $U$ be an open subset of $ \mathbb R^n$ and consider  $f :\mathbb R^n \to \mathbb R$ with the properties that  $ f( \partial U)=0$ and $f$ takes negative values on $U$. My questions:


*

*Is there any name for such functions in Mathematics Literature?

*Given any set $U$ as above can we always find a continuous function with above properties?

*Are these functions important at all? 
Thank you for your help.
 A: There's an old theorem (due to Whitney, I think) that says the following:

Given any smooth manifold $M$ and any closed subset $K\subseteq M$, there exists a smooth function $f\colon M\to \mathbb R$ that satisfies $f=0$ on $K$ and $f>0$ on $M\smallsetminus K$.

You can find a proof in my Introduction to Smooth Manifolds (2nd ed.), Theorem 2.29.
In your situation, if you apply this with $K=\mathbb R^n\smallsetminus U$ and then take the negative of the resulting function, you get a smooth solution to question 2.
Is there a name for such functions? They might sometimes be called (negative) bump functions, but that term more commonly is applied to a function whose zero set contains a given closed set (or equivalently, whose support is contained in some given open set). In the special case that $\partial U$ is a smooth hypersurface, a smooth function $f$ such that $f<0$ (or $>0$) on $U$, $f^{-1}(0) = \partial U$, and $df\ne 0$ on $\partial U$ is called a defining function for $U$.
As to your third question, bump functions and defining functions are extremely useful throughout differential geometry and PDE theory. But functions of the type guaranteed by the Whitney theorem I quoted above seem to be mainly curiosities. They provide an answer to the question "How special are the zero sets of smooth functions?" (Answer: not special at all, except for the fact that they're closed.)
