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categories: Re: stupid question?
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.
Computer Science Department
California State University, Fresno