From: Todd Wilson <twilson@csufresno.edu>
To: categories@mta.ca
Subject: Re: stupid question?
Date: Wed, 29 Mar 2000 19:00:25 -0800 (PST) [thread overview]
Message-ID: <14562.49993.312683.542317@goedel.engr.csufresno.edu> (raw)
In-Reply-To: <200003292313.SAA14643@hans.math.upenn.edu>
On Wed, 29 Mar 2000, Max Kanovitch wrote:
> The real fun is about a function f such that
> f is unbounded in any open interval (c,d), and
> in addition to that: f(x+y) = f(x)+f(y).
Using AC/Zorn's Lemma, we can construct 2^(2^Aleph_0) such functions,
as follows. Let B be a basis for the reals R as a rational vector
space. Clearly, |B| = 2^Aleph_0. For any non-empty proper subset C
of B, let g be its characteristic function (g(x)=1 if x in C, g(x)=0
otherwise) and let f be the unique linear extension of g to R. Then f
is linear, and its graph is dense in R^2, since, if c and d are such
that g(c)=1 and g(d)=0, then f(qc+rd) = q for all rationals q,r, and r
can be varied to make qc+rd as close as desired to any given real.
--
Todd Wilson
Computer Science Department
California State University, Fresno
next prev parent reply other threads:[~2000-03-30 3:00 UTC|newest]
Thread overview: 5+ messages / expand[flat|nested] mbox.gz Atom feed top
2000-03-29 23:13 Max Kanovitch
2000-03-30 3:00 ` Todd Wilson [this message]
-- strict thread matches above, loose matches on Subject: below --
2000-03-29 21:48 Wendt, Michael - SSMD/DMES
2000-03-29 20:23 Peter Freyd
2000-03-29 17:36 M.M. Mawanda
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