Derivation/Linear/Example

Let ${\displaystyle {}C^{0}(\mathbb {R} ,\mathbb {R} )}$ denote the space of continuous functions from ${\displaystyle {}\mathbb {R} }$ to ${\displaystyle {}\mathbb {R} }$, and let ${\displaystyle {}C^{1}(\mathbb {R} ,\mathbb {R} )}$ denote the space of continuously differentiable functions. Then the mapping

${\displaystyle D\colon C^{1}(\mathbb {R} ,\mathbb {R} )\longrightarrow C^{0}(\mathbb {R} ,\mathbb {R} ),f\longmapsto f',}$

which assigns to a function its derivative, is linear. In analysis, we prove that

${\displaystyle {}(af+bg)'=af'+bg'\,}$

holds for ${\displaystyle {}a,b\in \mathbb {R} }$ and another function ${\displaystyle {}g\in C^{1}(\mathbb {R} ,\mathbb {R} )}$.