Lebl-Basic-Analysis-I · Problem 4.1.14

The Derivative

Mathematics · differentiability-linear-approximation

Problem

Suppose

f : I \to \mathbb{R}

is differentiable at

c \in I

. Prove that there exist numbers

a

and

b

with the property that for every

\epsilon > 0

, there is a

\delta > 0

, such that

\bigl|a + b(x-c) - f(x)\bigr| \leq \epsilon\,|x-c|

, whenever

x \in I

and

|x-c| < \delta

. In other words, show that there exists a function

g : I \to \mathbb{R}

such that

\lim_{x \to c} g(x) = 0

and

\bigl|a + b(x-c) - f(x)\bigr| = g(x)|x-c|