Differentiable function/D in R/Linear approximation/Fact

Let ${}D\subseteq \mathbb {R}$ be a subset, ${}a\in D$ a point, and

$f\colon D\longrightarrow \mathbb {R}$

a function.

Then ${}f$ is

differentiable in ${}a$ if and only if there exists some ${}s\in \mathbb {R}$ and a function

$r\colon D\longrightarrow \mathbb {R} ,$

such that ${}r$ is continuous in ${}a$, ${}r(a)=0$, and such that

${}f(x)=f(a)+s\cdot (x-a)+r(x)(x-a)\,.$

Proof, Write another proof