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From: matt@oddjob.UChicago.UUCP (Matt Crawford)
Newsgroups: net.math
Subject: Re: f(x) = (if x = p/q then 1/q else 0) integrable ??
Message-ID: <589@oddjob.UChicago.UUCP>
Date: Wed, 30-Jan-85 12:40:10 EST
Article-I.D.: oddjob.589
Posted: Wed Jan 30 12:40:10 1985
Date-Received: Thu, 31-Jan-85 02:31:13 EST
References: <350@decwrl.UUCP> <ucbvax.4467>
Reply-To: matt@oddjob.UUCP (Matt Crawford)
Organization: U. Chicago: Astronomy & Astrophysics
Lines: 10

All you people who say "yes, it is integrable ..." may be deceiving
some readers.  The *Riemann* integral exists only if the limit of
the finite sums exists as the `mesh size' (largest difference between
any two mesh points) goes to zero.  I can always choose a mesh of
all rational or all irrational points and get different answers.
Therefore, the function is not *Riemann* integrable.  And in the
words of Richard Feynman:  "Lebesgue, Schlemesgue!"
_____________________________________________________
Matt		University	crawford@anl-mcs.arpa
Crawford	of Chicago	ihnp4!oddjob!matt