# Differentiable function/D in R/Linear approximation/Fact/Proof

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

$cases}"): {\displaystyle {{}} r(x) = \begin{cases} \frac{ f (x )-f (a) }{ x -a } - s \text{ for } x \neq a\, , \\ 0 \text{ for } x [[Category:Wikiversity soft redirects|Differentiable function/D in R/Linear approximation/Fact/Proof]] __NOINDEX__ a \, . \end{cases} \,$

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.