Derivative Operator is Linear Mapping

Theorem
Let $I := \closedint a b$ be a closed real interval.

Let $C \closedint a b$ be the space of real-valued functions continuous on $I$.

Let $C^1 \closedint a b$ be the space of real-valued functions continuously differentiable on $I$.

Let $D$ be the derivative operator such that:


 * $D : \map {C^1} I \to \map C I$

and $Dx := x'$.

Then $D$ is a linear mapping.

Positive homogenity
By definition, $D$ is a linear mapping