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!]
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?
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.