Integral of Compactly Supported Function

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

Let $K \subset \R$ be a compact subset, say, $\closedint a b$.

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


 * $\map \supp f = K$.

Then:


 * $\ds \int_{- \infty}^\infty \map f x \rd x = \int_a^b \map f x \rd x$