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From: springer@iuvax.UUCP
Newsgroups: net.math
Subject: Re: uniform continuity
Message-ID: <7000004@iuvax.UUCP>
Date: Tue, 29-Jan-85 10:04:00 EST
Article-I.D.: iuvax.7000004
Posted: Tue Jan 29 10:04:00 1985
Date-Received: Thu, 31-Jan-85 06:41:06 EST
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Nf-From: iuvax!springer    Jan 29 10:04:00 1985


The equivalent definition that you propose works only when the function
f is define on an arcwise connected set.  A function can be uniformly
continuous on a set made up any number of disconnected pieces.  In such
a case, you cannot relate the values in the different pieces the way
your formulation does.  However, the correct theorem which says that
a function f is uniformly continuous on a connected set S if and only if
given any epsilon > 0, there exists a finite number K such that for any
points x and y in S, |f(x) - f(y)| < K|x-y| + epsilon, is an interesting
fact and would actually make a nice exercise in an advanced calculus
textbook.
	....George Springer
            Indiana University