Lebl-Basic-Analysis-I · Problem 4.4.1

The Derivative

Mathematics · inverse-function-theorem-global

Problem

Suppose

f : \mathbb{R} \to \mathbb{R}

is continuously differentiable and

f'(x) > 0

for all

x

. Show that

f

is invertible on the interval

J = f(\mathbb{R})

, the inverse is continuously differentiable, and

(f^{-1})'(y) > 0

for all

y \in f(\mathbb{R})