Continuous Non-Negative Real Function with Zero Integral is Zero Function

Theorem
Let $a, b$ be real numbers with $a < b$.

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

Let:


 * $\map f x \ge 0$

for all $x \in \closedint a b$.

Let:


 * $\ds \int_a^b \map f x \rd x = 0$

Then $\map f x = 0$ for all $x \in \closedint a b$.