Why is the support defined as a closure?

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?

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.

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