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Math Help - Amazing Theorem? - Please help

  1. #1
    Super Member Bernhard's Avatar
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    Amazing Theorem? - Please help

    On page 252 of William Dunham's "Journey Through Genius" he is writing about Nineteenth Century mathematics and writes the following:

    "As the nineteenth Century progressed, mathematical discoveries came to light indicating that these two classes of numbers [rationals and irrationals] did not carry the same weight. The discoveries often required very technical, very subtle reasoning. For instance a function was described that was continuous at each irrational point and discontinuous at each rational point; however it was also proved that no function exists that is continuous at each rational point and discontinuous at each irrational point ... ... "

    Can anyone give me more information about this function and theorem(s) involved and point me to a text or reference where a proof is given.

    [I thought for a moment that the reference might be to DIrichlet's Charactersitic Function of the Rationals where f(x) is defined as 1 if x is rational and 0 if x is not rational - but David Bressoud in A Radical Approach to Lebesgue's Theory of Integration states on page 45 that "Dirichlet's function is totally discontinous since it is discontinous at every point!]
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Bernhard View Post
    On page 252 of William Dunham's "Journey Through Genius" he is writing about Nineteenth Century mathematics and writes the following:

    "As the nineteenth Century progressed, mathematical discoveries came to light indicating that these two classes of numbers [rationals and irrationals] did not carry the same weight. The discoveries often required very technical, very subtle reasoning. For instance a function was described that was continuous at each irrational point and discontinuous at each rational point; however it was also proved that no function exists that is continuous at each rational point and discontinuous at each irrational point ... ... "

    Can anyone give me more information about this function and theorem(s) involved and point me to a text or reference where a proof is given.

    [I thought for a moment that the reference might be to DIrichlet's Charactersitic Function of the Rationals where f(x) is defined as 1 if x is rational and 0 if x is not rational - but David Bressoud in A Radical Approach to Lebesgue's Theory of Integration states on page 45 that "Dirichlet's function is totally discontinous since it is discontinous at every point!]
    The relevant theorem says that if f:\mathbb{R}\to\mathbb{R} then it's set of discontinuities is a F_{\sigma} and thus a meager set. Thus, since \mathbb{Q} is a meager set so would the union of \mathbb{Q} and the irrationals, which is \mathbb{R}. But isn't \mathbb{R} a Baire space?
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    Super Member Bernhard's Avatar
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    Thanks, but can you help further ...

    Thanks for the help ... but can you please help further ...

    Can you recommend a good text that covers this theorem and its proof.

    Is there such a text accessible to undergraduates?

    Bernhard
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Bernhard View Post
    Thanks for the help ... but can you please help further ...

    Can you recommend a good text that covers this theorem and its proof.

    Is there such a text accessible to undergraduates?

    Bernhard
    Oh God, I don't know. I only tangentially know about this from other stuff. I would guess that Rudin's Real and Complex Analysis or Royden's book would be a good start. I would wait for someone more used to this subject to give you a good suggestion for a book.

    What year undergraduate are you? Those two books are in general graduate texts.

    Until then I managed to find the exact topic on wikipedia

    See here, here, and here.
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    Super Member Bernhard's Avatar
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    Thanks for the Wikipedia refs

    Thanks for the Wikipedia links ... ... most helpful.

    Also found a brief discussion on Thomae's Function in Stephen Abbotts undergraduater analysis book, "Undertanding Analysis"

    I am not taking a formal course in maths but am a math hobbyist - but my level is about senior undergraduate.

    Bernhard
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  6. #6
    Senior Member roninpro's Avatar
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    Hello.

    Consider the following function:

    <br />
     f(x)=\begin{cases}<br /> <br />
\frac{1}{n} & \text{for } x=\frac{m}{n} \text{ rational, in lowest terms} \\<br /> <br />
 0 & \text{for } x \text{ irrational}<br />
\end{cases}<br />

    You can can show that f is continuous at the irrationals and discontinuous on the rationals.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by roninpro View Post
    Hello.

    Consider the following function:

    <br />
     f(x)=\begin{cases}<br /> <br />
\frac{1}{n} & \text{for } x=\frac{m}{n} \text{ rational, in lowest terms} \\<br /> <br />
 0 & \text{for } x \text{ irrational}<br />
\end{cases}<br />

    You can can show that f is continuous at the irrationals and discontinuous on the rationals.
    Yes, that is Thomae's function. He is asking for a function whose set of continuities is the rationals and discontinuities is the irrationals.
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