My first thought was that any discontinuous linear function has a graph that's dense in R2, but since their construction requires the axiom of choice, that's probably not what you're looking for But you can use the same construction, just stopping after countably many steps, to get the same effect. In fact, I think just one step suffices. Define f as If x = a + b sqrt(2), a, b, rational, then f(x) = a + 2b. otherwise, f(x) = 0. finding a, b, where f(x) ~ y means finding a, b where a + b sqrt (2) ~ x, and a+ 2b ~ y Solving these two linear equations gives the desired values of a and b. They won't be rational, but choosing a' and b' arbitrarily close to a and b will give a point in the graph of the function arbitrarily close to (x, y). Andy On Tue, Apr 14, 2020 at 12:32 PM Allan Wechsler <acwacw@gmail.com> wrote:
Intuitively, there ought to be a R->R function whose graph is dense in R^2. But I haven't been able to come up with one quickly. Is there a classic example? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com