Let if and if . Show that is discontinuous for all
I know that I should find a sequence where , where is rational for even and irrational for odd (or vice versa), but I do not know how to find such a sequence.
But that is for specific values of , is there any way to have a generalized case? Because the general idea and hint that the book suggests is that when you have be rational for even and irrational for odd , so that the sequence does not converge at all. Because to me, it seems that all you've shown is that the function is discontinuous for irrational (because I believe there is a lemma that states that any real number is the limit of a sequence of rational numbers).
Edit: Nevermind, if we have , then , but all the terms of the sequence are irrational.
Thanks. I know it's not the topic, but could you perhaps prove that? We never proved in my class that any real number is the limit of a sequence of rationals or irrationals.
Edit: D'oh, I guess the sequence I suggested in my previous post shows the irrational part of the proof, but how is the rational part proven?
Let since both are dense in there exists sequences of points which lie in both. Now, if we assume that is continuous at then . Contradiction.
Alternatively, assume that is continuous at and WLOG assume that . Clearly then but by continuity we should be able to find some such that . See a problem?