Basic Analysis I — Real Analysis · Problem 4.3.9

The Derivative — Jiri Lebl

Mathematics — Real Analysis · extension-bounded-second-derivative

Problem

Suppose

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

is differentiable and

f''

is bounded. a) Show that there exists a once differentiable function

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

such that

f(x) = g(x)

for all

x \neq 0

. Hint: See Exercise 4.2.14. b) Find an example where the

g

is not twice differentiable at

x = 0