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.