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From: vdb@hou2g.UUCP (R.VANDERBEI)
Newsgroups: net.math
Subject: uniform continuity
Message-ID: <393@hou2g.UUCP>
Date: Sun, 27-Jan-85 13:13:42 EST
Article-I.D.: hou2g.393
Posted: Sun Jan 27 13:13:42 1985
Date-Received: Mon, 28-Jan-85 06:07:36 EST
Organization: AT&T Bell Labs, Holmdel NJ
Lines: 28

The usual definition of uniform continuity goes as follows:

     for every  epsilon > 0,  there exists a  delta > 0 such that
     |f(x) - f(y)| < epsilon  whenever  |x - y| < delta.

It is not hard (and so I leave it to you readers) to show that
the following condition is equivalent:

     for every  epsilon > 0, there exists a  K < infinity  such that
     for all x and y,  |f(x) - f(y)| < K |x - y| + epsilon.

This second formulation shows that uniform continuity is almost 
the same as Lipschitz continuity (which corresponds to being able
to find a finite  K  even when  epsilon is zero).

If you think about it (and especially, if you write down a proof),
you will see that the first condition is like a  "differential"
statement and the second condition is like an "integral" version.

The reason I was interested in this is because I wanted to know
whether a uniformly continuous function could grow more rapidly
than linearly at infinity.  The second condition shows that it 
cannot.

The second condition appears so simple and ellegant (and the proof is
also straight forward) that you'd expect to be able to find it
in any textbook on analysis however I have not seen it.  Does any
one know a reference?