I have two homework problems that I was hoping someone could help me with:
1. Prove that f(x)= sin(1/x), where x can't equal zero and f(0)=0 is not continuous at x=0 by finding an ε for which there is no reply.
2. At what values of x is
f(x) = (piecewise defined) 0, if x is irrational or = sinx if x is rational
For the first one see this thread
For the second one, my guess would be that it's continous at all points of the form where since for these points (and only for these).
And to prove it's continuous nowhere else, let . Since , there exists an irrational number , . So choosing ensures that is not continuous at .
A similar argument can be constructed if is irrational.
For what it is worth, another solution to 1.
You know that when where ,
Lets define a sequence where .
Also the define the sequence , where .
It is clear that
So we will substitude the sequences for
Now we have two sequences and with the same limit as 1/x such that and have two different limits.
Hence does not exist, so does not exist either.
Of course you are correct, I wonder why I didnt notice the error?
Any way corrected version above.