inverse functions

increasing function

given

A \subseteq ℝ

, a function

f : A \to ℝ

is increasing if given

x, y \in A

with

x < y

we have

f (x) \leq f (y)

. An increasing function is also known as a non-decreasing function.

strictly increasing function

given

A \subseteq ℝ

, a function

f : A \to ℝ

is strictly increasing if given

x, y \in A

with

x < y

we have

f (x) < f (y)

decreasing function

given

A \subseteq ℝ

, a function

f : A \to ℝ

is decreasing if given

x, y \in A

with

x < y

we have

f (x) \geq f (y)

. A decreasing function is also known as a non-increasing function.

strictly decreasing function

given

A \subseteq ℝ

, a function

f : A \to ℝ

is decreasing if given

x, y \in A

with

x < y

we have

f (x) > f (y)

monotone function

a function defined on a subset of

ℝ

is said to be monotone if and only if it is non-increasing or non-decreasing

strictly monotone function

a function defined on a subset of

ℝ

is said to be monotone if and only if it is stictly increasing or strictly decreasing

invertible function

Let

f : X \to Y

be a function, if there is a function

g : Y \to X

such that

g (f (x)) = x

for all

x \in X

and

f (g (y)) = y

for all

y \in Y

a function is invertible iff it is bijective

TODO

f

is strictly monotone then it is invertible

TODO

Suppose that

f : X \to Y

is strictly monotone, then

f^{- 1} : Y \to X

is a continuous function

Suppose without loss of generality that $f$ is strictly increasing on $I = (a, b)$ .

We'll show that the function is continuous, so let $p \in I$ and we'll show that $\lim_{y \to p} f^{- 1} (x) = f^{- 1} (p)$ using the epsilon delta definition.

Let $ϵ \in ℝ^{> 0}$ , let $x = f^{- 1} (p)$ and note that $x - ϵ < x + ϵ$ and therefore since $f$ is strictly increasing we know that $f (x - ϵ) < f (x + ϵ)$ , in other words, there is some $δ_{1}, δ_{2} \in ℝ^{> 0}$ such that $f (x - ϵ) = p - δ_{1}$ and $f (x + ϵ) = p + δ_{2}$ , now take $δ = \min (δ_{1}, δ_{2})$ and suppose $| y - p | < δ$ .

if $| y - p | < δ$ , then $δ - p < y < δ + p$