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From: ags@pucc-i (Dave Seaman)
Newsgroups: net.math
Subject: Re: f(x) = (if x = p/q then 1/q else 0) integrable ??
Message-ID: <871@pucc-i>
Date: Wed, 30-Jan-85 10:04:54 EST
Article-I.D.: pucc-i.871
Posted: Wed Jan 30 10:04:54 1985
Date-Received: Thu, 31-Jan-85 07:26:48 EST
References: <613@spuxll.UUCP>
Organization: Purdue University Computing Center
Lines: 16

>  Sorry about that: you have to leave 0 out of the integration range, since
>  f(0) is undefined.  Presuming 0 is not in the range, the integral is
>  indeed 0.

In the first place, f(0) is perfectly well-defined.  Since 0 = 0/1, and
since gcd(0,1) = 1 (which is what it means to say that a fraction is in
lowest terms), it follows that f(0) = 1.  (You do have to be somewhat more
precise about the definition and require q>0, but that applies to all
p/q, not only at zero).

Second, whether f is defined at zero has nothing to do with whether
the integral exists.  The function could be undefined even at uncountably
many points (the Cantor set, for example) and still have an integral.
It is only necessary that f be defined almost everywhere.
-- 
Dave Seaman			 ..!pur-ee!pucc-i:ags