Amazing Theorem? - Please help

Jan 2010
594
5
Hobart, Tasmania, Australia
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!]
 

Drexel28

MHF Hall of Honor
Nov 2009
4,563
1,566
Berkeley, California
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 \(\displaystyle f:\mathbb{R}\to\mathbb{R}\) then it's set of discontinuities is a \(\displaystyle F_{\sigma}\) and thus a meager set. Thus, since \(\displaystyle \mathbb{Q}\) is a meager set so would the union of \(\displaystyle \mathbb{Q}\) and the irrationals, which is \(\displaystyle \mathbb{R}\). But isn't \(\displaystyle \mathbb{R}\) a Baire space?
 
Jan 2010
594
5
Hobart, Tasmania, Australia
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
 

Drexel28

MHF Hall of Honor
Nov 2009
4,563
1,566
Berkeley, California
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.
 
Jan 2010
594
5
Hobart, Tasmania, Australia
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
 
Nov 2009
485
184
Hello.

Consider the following function:

\(\displaystyle
f(x)=\begin{cases}

\frac{1}{n} & \text{for } x=\frac{m}{n} \text{ rational, in lowest terms} \\

0 & \text{for } x \text{ irrational}
\end{cases}
\)

You can can show that \(\displaystyle f\) is continuous at the irrationals and discontinuous on the rationals.
 

Drexel28

MHF Hall of Honor
Nov 2009
4,563
1,566
Berkeley, California
Hello.

Consider the following function:

\(\displaystyle
f(x)=\begin{cases}

\frac{1}{n} & \text{for } x=\frac{m}{n} \text{ rational, in lowest terms} \\

0 & \text{for } x \text{ irrational}
\end{cases}
\)

You can can show that \(\displaystyle f\) 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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