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From: bstewart@bnl.UUCP (Hugh Bruce Stewart)
Newsgroups: net.math
Subject: Re: Re: Re: Non-linear systems: disconti
Message-ID: <876@bnl.UUCP>
Date: Tue, 12-Feb-85 23:26:47 EST
Article-I.D.: bnl.876
Posted: Tue Feb 12 23:26:47 1985
Date-Received: Mon, 18-Feb-85 04:17:07 EST
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> 
> Speaking of discontinous functions ...
> 
> One of my favorite functions is
> 
> 		{ 1/q,	x rational and expressed as p/q in lowest terms
> 	 f(x) = {
> 		{ 0,	x irrational
> 

  Anyone with an interest in such things might want to look at the
book Counterexamples in Analysis by Gelbaum and Olmstead, Holden-Day,
1964.  This is a unique compendium of all the things that can go 
wrong if theorems about continuity, integration, convergence, etc.
are misused.
  Chapter 8 in particular shows why Lebesgues's definition of a
definite integral is useful. One might say that his definition
is made to handle functions such as density distributions on
deterministically defined fractal sets. In a nutshell, Lebesgue's
idea was to approximate the area under the curve f(x) by
summing over the f(x)-axis instead of over the x-axis as
Cauchy and Riemann had done.
 
Bruce Stewart, Applied Math. Dept., Brookhaven National Lab.