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

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

a function. Then ${\displaystyle {}f}$ is differentiable in ${\displaystyle {}a}$ if and only if there exists some ${\displaystyle {}s\in \mathbb {R} }$ and a function

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

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

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

Proof

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

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

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

$cases}"): {\displaystyle {{}} r(x) = \begin{cases} \frac{ f (x )-f (a) }{ x -a } - s \text{ for } x \neq a\, , \\ 0 \text{ for } x [[Category:Wikiversity soft redirects|Real functions/Differentiability/Linear approximation/Section]] __NOINDEX__ a \, . \end{cases} \,$

Because of differentiability, the limit

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

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

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

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

${\displaystyle \Box }$

The affine-linear function

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

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

Corollary

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

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

a function. Then ${\displaystyle {}f}$ is also continuous in ${\displaystyle {}a}$.

Proof

This follows immediately from

${\displaystyle \Box }$