Lebl-Basic-Analysis-I · Problem 4.4.8

The Derivative

Mathematics · inverse-function-differentiability

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 := 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

.

Related theory

Smooth Manifolds And The Inverse Function Theorem

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