I have never liked problems of the sort where rational/irrational numbers are involved.
For the function
determine where is continuous and where is discontinuous. Does have any partial derivatives?
A note provided is as follows: Every neighborhood in always contains points with rational and points with irrational coordinates.
Mainly, the part I have difficulty with is the rational/irrational bit. I've never been able to work with those properly, probably because I missed something fundamental in my earlier years.