I need to find when (where ).Prove that is discontinuous at .
Suppose that is rational and is rational. Therefore:
This is valid from the definition of .
Suppose that is rational and is irrational.
Hence take .
So the proof for when is rational is as follows:
Let , be rational and . When I have:
Suppose that is irrational and is also irrational.
Therefore pick as before.
Suppose that is irrational and is rational.
At this point, a value cannot be chosen since it isn't in the above inequality chain. Since has to be valid , there is a clear contradiction (eg, if I can make ).
Is this sufficient proof to show that is discontinuous at ?
(btw, I have shown that is continuous at
PS. Can someone show me a way of doing this with sequences? This method is a little long.