Countable Sets

Discussion in 'Undergraduate Math' started by SHEN, Jan 24, 2006.

  1. SHEN

    SHEN Guest

    Hi, everyone,

    Another difficulty to prove the following set to be countable,

    1. f(x) is a real function defined on R^1.
    2. for all x0 in R^1, exists delta>0, f(x)>=f(x0) holds when |x-x0|<delta.
    Prove: f(R^1) is countable.

    SHEN
     
    SHEN, Jan 24, 2006
    #1
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  2. On Tue, 24 Jan 2006 22:31:56 +0800, SHEN
    Let S = f(R^1), and suppose that S is uncountable. For each
    y in S choose x(y) in R^1 such that y = f(x(y)), and let A =
    {x(y) : y in S}. Show that for each x in A there are
    rational numbers p(x) and q(x) such that p(x) < x < q(x),
    and f(u) >= f(x) whenever p(x) < u < q(x). Conclude that
    there are distinct x and x' in A such that p(x) = p(x') and
    q(x) = q(x'). Finally, consider f(x) and f(x') to get a
    contradiction.

    Brian
     
    Brian M. Scott, Jan 24, 2006
    #2
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  3. On Tue, 24 Jan 2006 17:20:52 -0500, "Brian M. Scott"
    By the way, the converse is also true, and a nice little
    problem. Let S be any countable set of real numbers. Then
    there is a function f satisfying your conditions (1) and (2)
    and having the further property that f(R^1) = S.

    Brian
     
    Brian M. Scott, Jan 25, 2006
    #3
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