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coffeeking

Member

Registered: 2007-11-18

Posts: 44

Uniform Continuity

Which of these functions (or both) is/are uniformly continuous and why?

Sorry if my question sound noobish... Thanks in advance.:D

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Uniform Continuity

The definition of “uniformly continuous” is:

For the first question

is not uniformly continuous. To prove this, you need to find an

with the property that for all

, there exist complex numbers

and

such that

but

.

Well, I can tell you that taking

will work. Given

, take

Then

but

I leave you to do the second one yourself.

Last edited by JaneFairfax (2009-03-02 06:43:14)

coffeeking

Member

Registered: 2007-11-18

Posts: 44

Re: Uniform Continuity

Hi Jane, thanks for helping, I appreciate it. Well, I do agree with you that I should feel ashamed of myself for asking such a simple question, in fact I already did ...

Actually even since my school term started I have met a lot of difficulties trying to understand this course as my lecture notes are really wordy, lack of illustration and worked examples and full of typo errors. Furthermore as to make the matter worse, my lecturer is not being clear in his explanation. In fact, a lot of his students including me feel even more doubtful after asking him question in the hope that he could clear our doubts.

Currently I have borrowed lecture notes from my friends from other university, which seem to be 100 times better than mine and I have been trying my very best to catch up. I became more and more frustrated after looking through my friends' notes and realised my lecturer has left out a great deal of definitions and illustrations in his notes. An example is that he did not even explain to us what is a simply connected domain when Cauchy's Integral Theorem was being taught.

Frankly speaking, when I was posting this question I am actually finding "the easy way out" as I have too much stuffs to catch up lately. I was hoping that I could in fact learn from the working posted in the forum as I find it easier to learn through worked example where my lecture notes really lacks.

I can say that I have learned something new today and I have to really thanks you again for your willingness to help and don't worry, I understand your working.:P

Well, as for the second question I think I have figured out how to do. Below is my solution please check my working for me:

Suppose 

for

then       

             

             

             

Therefore choose

Hence

is uniformly continuous.

In addition I have another question that I am unsure of. My question is is it true that epsilon delta method when use to prove continuity we can only prove continuity at a point and not the whole function unlike for uniform continuity?

Thanks in advance.