Riemann Integral Operator is Continuous Linear Transformation

Theorem
Let $\struct {C \closedint a b, \norm {\, \cdot \,}_\infty}$ be the normed vector space of real-valued functions continuous on $\closedint a b \subseteq \R$ equipped with the supremum norm.

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


 * $\ds \forall \mathbf x \in C \closedint a b : \map T {\mathbf x} = \int_a^b \map {\mathbf x} t \rd t$

Then $T$ is a continuous mapping.

Proof
We have that Integral Operator is Linear.

Furthermore:

By Continuity of Linear Transformation between Normed Vector Spaces, $T$ is continuous.