I think you have a typo in your question.
Note if you get
This limit does not exists.
I think you mean the function
-Challenging question-
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.
Approved CB
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.