An interesting function
Let define on all real numbers and
Show that is nowhere continuous.
I suppose I was sent the solution so that I could approve this, but you are relying on me and/or the other senior staff being psychic rather than the more usual psychotic which as Sir Humphry would say is "Brave". In future try saying why you have sent such material.
I think you have a typo in your question.
Note if you get
This limit does not exists.
I think you mean the function
No typo indeed. And your deduction is part of the proof - the function is not continuous at x=1. Below is a valid proof, not very rigorous though.
Let . If is a rational number other than , that is, it can be rewritten in the form , where and . Once increases to become greater than , it forces .
On the other side, if is irrational, will approach . Hence given any real number , the limits do not agree when approach from irrationals and rationals respectively so the function is almost nowhere continuous and only proving the case is left.
In the case of , the limit does not exists, so the function is nowhere continuous.
See [Example 7.4, 1] for the solution.
Originally Posted by TerenceCS
Probably,  is the book which most of the MHF members read. (Happy)
 W. Rudin, Principles of Mathematical Analysis, 3rd Ed., McGraw-Hill Book Co., New York, 1976.