Uniform Continuous

Definition (Uniform continuous).

Let A be a non-empty subset of and let be a function
then is said to be uniform continuous if for every $ε$ there exist such that
$ε$ for all , with

Remark.

Here is not dependent on or where as in continuous functions continuous functions depends on

Clearly every uniform continuous functions is continuous

Example

Consider defined by

We need to estimate

Take $ε$ then for we have $ε$ so is uniformly continuity
Consider

Claim.

is not Uniformly Continuous Uniformly Continuous

Take $ε$ suppose there exist such that

take with

we shall have for

We shall have

Take then we have so above above dosen't hold for

Theorem.

Let with Let be a continuous function. Then is uniformly continuous

Proof.
Suppose is not uniformly continuous we will arrive at a contradiction
if is not uniformly continuous then there exist $ε$ for which no
satisfies

ε

So for taking there exist
such that and $ε$

By bolzano-Weierstrass theorem bolzano-Weierstrass theorem has a convergent subsequent
say converges to some

We have

take and for

converges to and

We also have

ε

$*$ imply

ε

We have taken so and converges to the same value
hence
but we have

ε

So are not same hence they are not continuous which contradicts that is continuous
Hence our assumption that is not uniformly Continuous is false

Theorem.

Let A be a non-empty subset of and let be uniform continuous. Suppose