Originally Posted by

**Runty** I have never liked problems of the sort where rational/irrational numbers are involved.

For the function $\displaystyle f(x,y)=\left\{\begin{array}{cc}x^2+y^2,&x,y\mbox{ both rational }\\0, & \mbox{ otherwise, }\end{array}\right$

determine where $\displaystyle f$ is continuous and where $\displaystyle f$ is discontinuous. Does $\displaystyle f$ have any partial derivatives?

A note provided is as follows: Every neighborhood in $\displaystyle R^2$ 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.