Basic Analysis I — Real Analysis · Problem 4.4.8

The Derivative — Jiri Lebl

Mathematics — Real Analysis · inverse-differentiability-from-inverse

Problem

Suppose

I, J

are intervals and a monotone onto

f : I \to J

has an inverse

g : J \to I

. Suppose

x \in I

and

y \mathrel{\mathop:}= f(x) \in J

, and that

g

is differentiable at

y

. Prove: a) If

g'(y) \neq 0

, then

f

is differentiable at

x

. b) If

g'(y) = 0

, then

f

is not differentiable at

x