Countable discontinuity

Discussion in 'Undergraduate Math' started by SHEN, Jan 19, 2006.

  1. SHEN

    SHEN Guest


    I'm trying to prove that f(x), being bounded in the interval (a, b) and
    sign-preserving (i.e. if f(x0)> or <0, then f is positive or negative in
    some neighbourhood), has a countable set of discontinuities.

    Could anyone give some hint?

    SHEN, Jan 19, 2006
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  2. Hi,
    With the above assumptions on f, your statement is
    (Consider f(x) = 1, if x is rational, and f(x)=2 else)

    Best wishes
    Torsten Hennig, Jan 19, 2006
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  3. SHEN

    SHEN Guest

    You're right, but I'm still wondering the intension of the problem's author.
    SHEN, Jan 19, 2006
  4. Well since the problem as you stated it is wrong, it's hard to see
    what that means.

    Possibly the problem as the author stated it is actually correct,
    and you're misinterpreting what something means in the statement.
    You could try posting a word-for-word statement of the problem,
    _exactly_ as it appears in whatever source this came from.

    (For example the statement would be true if f were monotone,
    and so I wonder whether something that says that f is monotone
    got mangled somehow into your "sign-preserving" condition.
    One reason for wondering that is that I doubt that "sign-preserving"
    is the terminology the author used for "if f(x0)> or <0, then f
    is positive or negative in some neighbourhood", because
    that terminology for that condition doesn't make much sense.
    So I suspect you've rephrased some things...)


    David C. Ullrich
    David C. Ullrich, Jan 20, 2006
  5. SHEN

    SHEN Guest

    Sorry for late in response. Actually, it is the note of the author for
    the term "sign-preserving". I decide to give up this wrong one and go on
    the next.
    SHEN, Jan 24, 2006
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