Basic Analysis I — Real Analysis · Problem 4.4.6

The Derivative — Jiri Lebl

Mathematics — Real Analysis · inverse-function-theorem-failure

Problem

Let

f(x) \mathrel{\mathop:}= x + 2x^2 \sin(1/x)

for

x \neq 0

and

f(0) \mathrel{\mathop:}= 0

. Show that

f

is differentiable at all

x

, that

f'(0) > 0

, but that

f

is not invertible on any open interval containing the origin.