
# 6.1: Best Linear Approximations


## Definition

We say a function $$f: \mathbb{R} \rightarrow \mathbb{R}$$ is linear if for every $$x, y \in \mathbb{R}$$,

$f(x+y)=f(x)+f(y)$

and for every $$\alpha \in \mathbb{R}$$ and $$x \in \mathbb{R}$$,

$f(\alpha x)=\alpha f(x).$

## Exercise $$\PageIndex{1}$$

Show that if $$f: \mathbb{R} \rightarrow \mathbb{R}$$ is linear, then there exists $$m \in \mathbb{R}$$ such that $$f(x)=m x$$ for all $$x \in \mathbb{R}$$.

## Definition

Suppose $$D \in \mathbb{R}, f: D \rightarrow \mathbb{R},$$ and $$a$$ is an interior point of $$D$$. We say $$f$$ is differentiable at $$a$$ if there exists a linear function $$d f_{a}: \mathbb{R} \rightarrow \mathbb{R}$$ such that

$\lim _{x \rightarrow a} \frac{f(x)-f(a)-d f_{a}(x-a)}{x-a}=0.$

We call the function $$d f_{a}$$ the best linear approximation to $$f$$ at $$a,$$ or the differential of $$f$$ at $$a .$$

## Proposition $$\PageIndex{1}$$

Suppose $$D \subset \mathbb{R}, f: D \rightarrow \mathbb{R},$$ and $$a$$ is an interior point of $$D .$$ Then $$f$$ is differentiable at $$a$$ if and only if

$\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}$

exists, in which case $$d f_{a}(x)=m x$$ where

$m=\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}.$

Proof

Let $$m \in \mathbb{R}$$ and let $$L: \mathbb{R} \rightarrow \mathbb{R}$$ be the linear function $$L(x)=m x .$$ Then

\begin{aligned} \frac{f(x)-f(a)-L(x-a)}{x-a} &=\frac{f(x)-f(a)-m(x-a)}{x-a} \\ &=\frac{f(x)-f(a)}{x-a}-m. \end{aligned}

Hence

$\lim _{x \rightarrow a} \frac{f(x)-f(a)-L(x-a)}{x-a}=0$

if and only if

$\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}=m.$

Q.E.D.