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:

as required.

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.