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*To*: categories@mta.ca*Subject*: categories: Re: stupid question?*From*: Todd Wilson <twilson@csufresno.edu>*Date*: Wed, 29 Mar 2000 19:00:25 -0800 (PST)*In-Reply-To*: <200003292313.SAA14643@hans.math.upenn.edu>*References*: <200003292313.SAA14643@hans.math.upenn.edu>*Sender*: cat-dist@mta.ca

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

**References**:**categories: Re: Re: stupid question?***From:*maxkanov@math.upenn.edu (Max Kanovitch)

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