Basic Analysis I — Real Analysis · Problem 4.1.14

The Derivative — Jiri Lebl

Mathematics — Real Analysis · 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|