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

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

holds. Since ${}r$ is continuous in ${}a$, the limit on the left-hand side, for ${}x\rightarrow a$, exists.

$\Box$

The affine-linear function

$D\longrightarrow \mathbb {R} ,x\longmapsto f(a)+f'(a)(x-a),$

is called the affine-linear approximation. The constant function given by the value ${}f(a)$ can be considered as the constant approximation.

Corollary

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 also continuous in ${}a$.

Proof

This follows immediately from fact.

$\Box$