Lebl-Basic-Analysis-I · Problem 4.2.14

The Derivative

Mathematics · differentiability-extension

Problem

Suppose

f : (0,1) \to \mathbb{R}

is differentiable and

f'

is bounded. a) Show that there exists a continuous function

g : [0,1) \to \mathbb{R}

such that

f(x) = g(x)

for all

x \neq 0

. Hint: Proposition 3.4.6 and Exercise 4.2.3. b) Find an example where the

g

is not differentiable at

x = 0

. Hint: Consider something based on

\sin(\ln x)

, and assume you know basic properties of

\sin

and

\ln

from calculus. c) Instead of assuming that

f'

is bounded, assume that

\lim_{x \to 0} f'(x) = L

. Prove that not only does

g

exist but it is differentiable at

0

and

g'(0) = L