Riemann Integral Operator is Linear Mapping

Theorem
Let $C \closedint a b$ be the space of continuous Riemann integrable functions.

Let $\R$ be the set of real numbers.

Let $I : C \closedint a b \to \R$ be the Riemann integral operator.

Then $I$ is a linear mapping.

Proof
Let $x,y \in C \closedint a b$ be Riemann integrable.

Let $\alpha \in \R$.