Define f by
Then and certainly exist at every point except (0,0), and may be calculated by the quotient rule. They also exist at (0,0), although you need to use the definition of partial derivative to find them:
and may be found similarly to equal 0. However f is not differentiable at (0,0). In fact, f is not even continuous at (0,0)! To see this, approach the origin along the line y=x. Since f(x,x)=1/2 for all , the limit along this line is 1/2. But along the line x=0, f(0,y)=0 for , so along this line f approaches 1/2. Therefore the two dimensional limit doesn't exist. Here's a picture of f:
Although the partials of this function exist at every point, they can't be continuous everywhere, since there is a theorem telling us that functions with partial derivatives which are continuous in an open set must be differentiable in that set.