Basic Analysis I — Real Analysis · Problem 4.1.13

The Derivative — Jiri Lebl

Mathematics — Real Analysis · differentiability-square-of-function

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) \mathrel{\mathop:}= \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) \mathrel{\mathop:}= \bigl(f(x)\bigr)^2

is not differentiable at the origin.