Basic Analysis I — Real Analysis · Problem 4.4.7

The Derivative — Jiri Lebl

Mathematics — Real Analysis · inverse-function-global-conditions

Problem

a) Let

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

be a continuously differentiable function and

k > 0

be a number such that

f'(x) \geq k

for all

x \in \mathbb{R}

. Show

f

is one-to-one and onto, and has a continuously differentiable inverse

f^{-1} : \mathbb{R} \to \mathbb{R}

. b) Find an example

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

where

f'(x) > 0

for all

x

, but

f

is not onto.