Umm, I hate to disagree, but that example does not have partial derivatives at the origin:

$\displaystyle \frac{\partial f}{\partial x}(0,0) = \lim_{h\to0}\frac{f(h,0)-f(0,0)}h = \lim_{h\to0}\frac{1-0}h = $ (ouch!)

I think you probably meant $\displaystyle

f(x,y) = \begin{cases}1 &(xy\ne0),\\ 0&(x \text{ or }y =0).\end{cases}

$ But no, on second thoughts that won't work either, because that function doesn't have partial derivatives on either of the axes (except at the origin). I have a feeling that the example I gave previously can't really be simplified.