You are simply over-thinking this problem.
In any region of zero the numbers are themselves near zero.
So if .
Doing that simple step solves this problem.
I just want to make sure i'm doing these correctly.
I need to find when .Prove that is continuous at 0.
Let (since i'm interested in continuity at the origin).
Therefore I need to find when .
Let .
for .
(since ) for \ .
Therefore pick such that .
That's the preliminary work done, so here's the proof.
Let and define When :
for
for \
My one issue is that the method i've used is extremely similar to a method someone showed me how to use for another question. I also haven't used the properties of rational and irrational numbers as I would have liked.
Is this right?