TITLE: Why there are no constant functions on $\mathbb{R}$ with compact support?
QUESTION [5 upvotes]: For some the question might seem trivial but the concept is new to me and  I have been wondering 
why there are no constant functions on $\mathbb{R}$ with compact support?
Following wiki:
Def.1) Functions with compact support on a topological space $X$ are those whose support is a compact subset of $X$. 
Def.2) $\operatorname{supp}f=:\overline{\{x\in X\;,f(x)\neq 0\}}$
What if we take such function $f:\mathbb{R}\to\mathbb{R}$ defined as $f(x)=0\;,\forall x\in\mathbb{R}$ then $\operatorname{supp}f=\overline{\{x\in\mathbb{R}\;,f(x)\neq 0\}}=\overline{\emptyset}=\emptyset$ but $\emptyset$ is a compact subset of $\mathbb{R}$ I think....
Could someone enlighten me? Thank you.

REPLY [8 votes]: The author in the article you link to (www-personal.umich.edu/~wangzuoq/437W13/Notes/Lec%2030.pdf) is just being sloppy. He really should have said there are no non-zero constant functions with compact support on $\mathbb R$ (and thus $H_c^0(\mathbb R)$.
So yes, your example is correct. The empty set is (tautologically) compact.