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?

Bernhard

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