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about uni. continuous proof
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bill1158
science forum beginner


Joined: 09 Jul 2006
Posts: 12

PostPosted: Tue Jul 18, 2006 10:30 pm    Post subject: about uni. continuous proof Reply with quote

in proving uniform continuity, how do we choose the parameter d? any
helpful clues?
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Lynn Kurtz
science forum Guru


Joined: 02 May 2005
Posts: 603

PostPosted: Tue Jul 18, 2006 10:51 pm    Post subject: Re: about uni. continuous proof Reply with quote

On 18 Jul 2006 15:30:02 -0700, "bill" <bilgiaslankurt@gmail.com>
wrote:

Quote:
in proving uniform continuity, how do we choose the parameter d? any
helpful clues?

Your most helpful clues are likely to be in your textbook examples.
Given e > 0 you have to find a d, usually expressed in terms of e such
that:

| f(x) - f(y) | < e when |x - y| < e.

How you do that depends on the form of the function f(x) you are
dealing with. One common method involves using the mean value theorem
if you have the right conditions. If f' is continuous and | f'(x) | <
M on your domain you can use the MVT to write:

| f(x) - f(y) | < M | x - y |

which will be less than e if | x - y | < e / M

--Lynn
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