6.1: Best Linear Approximations
- Page ID
- 22671
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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 \(x \in \mathbb{R}\),
\[f(\alpha x)=\alpha f(x).\]
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 \(x \in \mathbb{R}\).
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 \(a .\)
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.