Lebl-Basic-Analysis-I · Problem 4.1.13

The Derivative

Mathematics · differentiability-composition

Problem

Suppose

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

is a function such that

f(x) = x\, h(x)

for a bounded function

h

. a) Show that

g(x) := \bigl(f(x)\bigr)^2

is differentiable at the origin and

g'(0) = 0

. b) Find an example of a continuous function

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

with

f(0) = 0

, but such that

g(x) := \bigl(f(x)\bigr)^2

is not differentiable at the origin.