TITLE: Why is the support defined as a closure?
QUESTION [20 upvotes]: In the definition of the support of a real function $f$ on $X$, why is it important to consider the closure of the set $S=\{x\in X:f(x)\neq0\}$ and not just $S$ itself? 
Why is the closure of $S$ called the "support" of $f$ or how did this name come about?

REPLY [13 votes]: An equivalent, and possibly more natural definition of the support is the following:  a point $x$ is in the support of $f$ precisely when $f$ does not vanish identically on a neighborhood of $x$. With this definition, it is obvious that the support is closed.
Vanishing on a neighborhood of $x$ is a much more significant property than simply vanishing at $x$, and for many purposes, it is the natural condition to consider.