Continuous Bijections

Definition (Increasing Decreasing Functions).

Let be a non empty subset of and be a functionn

i) is said to be increasing (non-decreasing) if
ii) is said to be strictly increasing if
iii) is said to be decreasing (non-increasing) if
iv) is said to be strictly decreasing if

Definition (Monotonic function).

If a function is increasing or decreasing it is said to be Monotonic

Definition (Strictly Monotonic function).

If a function is strictly increasing or strictly decreasing it is said to be strictly monotonic function

Theorem.

Let with let be a continuous function and
assume is a bijection
then is said strictly monotonic

Proof.

We claim or
Suppose
then there exist some
also there exist some such that
take then by Intermediate Value Theorem Intermediate Value Theorem
there exist such that

this contradicts the fact that is injective
So, either or

Claim.

if then and is strictly increasing

Assume that this is not true and apply Intermediate Value Theorem Intermediate Value Theorem
If is not strictly increasing as is not increasing we can get such that
there exist also there exist and such that hence it is not injective
similarly we can show

Claim.

if then and is strictly decreasing

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