Limit of function

Cluster point

Definition.

Let A be a non-empty subset of . Then is said to be cluster point of A is for every

we denote a set of cluster points of a by

Example

Theorem.

Let A be a non-empty subset of . Then is a cluster point of A iff there exist a sequence in converging to

Limit of function at a point

Definition.

Let A be a non-empty subset of and let be a function let be a cluster point of A
then is said to have a limit at if for every possible there exist such that
$ε$ for all
this is denoted by

Example

Theorem.

Let be a non-empty subset of and is a function. Suppose is a cluseter point of A, Then has limit at iff for every sequence in converging to , converges to

Proof.

Suppose the
then suppose is a sequence in converging to c

Claim.

converges to

For $ε$ there exist such that

ε

As converges to then there exist such that for
So, for
hence $ε$ for

Suppose does not converges to at then there exist $ε$

ε

False for all
in particularly for there exist

ε

then converges to but does not converge to

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Right limit

Definition.

Let A be a subset of then is said to be right cluster point of A if for all

Theorem.

TFAE
i) is a right cluster point of A
ii) there exist a sequence in converges to c
there exist a decreasing sequence in converging

Definition (Right Point Limit).

Let A be a non-empty subset of and let be a function
Let be a right cluster point A
then right hand side limit of at exist and equals is for every $ε$
$ε$

Notation: and

Left Limit

Definition.

Let A be a subset of then is said to be left cluster point of A if for all

Theorem.

TFAE
i) is a left cluster point of A
ii) there exist a sequence in converges to c
there exist a increasing sequence in converging

Definition (left Point Limit).

Let A be a non-empty subset of and let be a function
Let be a left cluster point A
then left hand side limit of at exist and equals is for every $ε$
$ε$

Notation: and

Theorem.

Let with left be an increasing function. Then for

prove this as an excercise

Theorem.

let be an increasing function then for

Proof.
Fix
take is defined as is non-empty and bounded by

for $ε$ $ε$ is not a lower bound
hence there exist with $ε$
as is increasing $ε$
Take then

ε

hence

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Theorem.

Let with and let be a continuous bijection then is continuous

Proof.

is strictly monotonic

case 1:
if is increasing

=> is increasing

similarly if is decreasing then is also decreasing

take from previous theorem

suppose
then

this shows that is not surjective

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