Value of Compactly Supported Function outside its Support

Theorem

Let $f : \R \to \R$ be a continuous real function.

Let $K \subseteq \R$ be a compact subset.

Let $K$ be the support of $f$:

$\map \supp f = K$.

Then:

$\forall x \notin K : \map f x = 0$

Proof

We have that:

$\R = K \cup \paren {\R \setminus K}$.

By definition of the support:

$x \in \map \supp f \iff \map f x \ne 0$

By Biconditional Equivalent to Biconditional of Negations:

$\neg \paren {x \in \map \supp f} \iff \neg \paren {\map f x \ne 0}$

That is:

$x \notin K \iff \map f x = 0$

or

$x \in \R \setminus K \iff \map f x = 0$

Hence:

$\forall x \notin K : \map f x = 0$

$\blacksquare$