Real functions/Differentiability/Linear approximation/Section

We discuss a property which is equivalent with differentiability, the existence of a linear approximation. This formulation is important in many respects: It allows giving quite simple proofs of the rules for differentiable functions, one can use it to reduce differentiability to the continuity of an error function, it yields a model for approximation with polynomials of higher degree (quadratic approximation, Taylor expansion), and it allows a direct generalization to the higher-dimensional situation.

Theorem

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

If ${}f$ is differentiable, then we set

{}s:=f'(a)\,.

Then the only possibility to fulfill the conditions for ${}r$ is

{}r(x)={\begin{cases}{\frac {f(x)-f(a)}{x-a}}-s{\text{ for }}x\neq a\,,\\0{\text{ for }}x=a\,.\end{cases}}\,

Because of differentiability, the limit

{}\operatorname {lim} _{x\rightarrow a,\,x\in D\setminus \{a\}}r(x)=\operatorname {lim} _{x\rightarrow a,\,x\in D\setminus \{a\}}{\left({\frac {f(x)-f(a)}{x-a}}-s\right)}\,

exists, and its value is ${}0$ . This means that ${}r$ is continuous in ${}a$ .
If ${}s$ and ${}r$ exist with the described properties, then for ${}x\neq a$ the relation