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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.

Todd Wilson
Computer Science Department
California State University, Fresno